Statement Exercises
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[{"Name":"Statement Exercises","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"2m 20s","ChapterTopicVideoID":12781,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12781.jpeg","UploadDate":"2018-08-16T06:10:07.7730000","DurationForVideoObject":"PT2M20S","Description":null,"MetaTitle":"Exercise 1 - Statement Exercises: Practice Makes Perfect | Proprep","MetaDescription":"Studied the topic name and want to practice? Here are some exercises on Statement Exercises practice questions for you to maximize your understanding.","Canonical":"https://www.proprep.uk/general-modules/all/probability/statement-exercises-_-probability/statement-exercises/vid13260","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"Events A and B are in a sense, space Omega."},{"Start":"00:02.415 ","End":"00:05.040","Text":"Now it\u0027s known that the probability of A equals to"},{"Start":"00:05.040 ","End":"00:08.340","Text":"the probability of B and that equals to 0.3."},{"Start":"00:08.340 ","End":"00:12.420","Text":"The statement asserts that the probability of exactly 1 event"},{"Start":"00:12.420 ","End":"00:17.955","Text":"occurring if the events are mutually exclusive is 0.42."},{"Start":"00:17.955 ","End":"00:22.515","Text":"Now, this statement is not true."},{"Start":"00:22.515 ","End":"00:26.360","Text":"Why is that? Well, let\u0027s take a look at what\u0027s going on here."},{"Start":"00:26.360 ","End":"00:29.000","Text":"Here we have our sample space Omega,"},{"Start":"00:29.000 ","End":"00:32.870","Text":"and here we have the 2 events and we see that they\u0027re mutually exclusive."},{"Start":"00:32.870 ","End":"00:35.480","Text":"Now, the probability of A that equals to"},{"Start":"00:35.480 ","End":"00:39.755","Text":"0.3 and the probability of B that also equals to 0.3."},{"Start":"00:39.755 ","End":"00:44.870","Text":"Now, we\u0027re asking for the probability of exactly 1 event occurring."},{"Start":"00:44.870 ","End":"00:50.165","Text":"That means that we\u0027re looking at the probability of A plus the probability of B."},{"Start":"00:50.165 ","End":"00:54.410","Text":"Now that equals to 0.3 plus 0.3,"},{"Start":"00:54.410 ","End":"01:00.605","Text":"and that equals to 0.6 which does not equal to 0.42."},{"Start":"01:00.605 ","End":"01:03.860","Text":"Now, there\u0027s another way of looking at this."},{"Start":"01:03.860 ","End":"01:06.980","Text":"Here we have our probability matrix now"},{"Start":"01:06.980 ","End":"01:10.910","Text":"that the sum of other probabilities that needs to be 1."},{"Start":"01:10.910 ","End":"01:15.770","Text":"Now we have that the events are mutually exclusive,"},{"Start":"01:15.770 ","End":"01:20.165","Text":"so the probability of B intersect that equals to 0."},{"Start":"01:20.165 ","End":"01:24.050","Text":"Now we have that the probability of A that equals to"},{"Start":"01:24.050 ","End":"01:28.700","Text":"0.3 and the probability of B that also equals to 0.3."},{"Start":"01:28.700 ","End":"01:31.655","Text":"Now having filled in what\u0027s given,"},{"Start":"01:31.655 ","End":"01:34.820","Text":"we can fill in the rest of this table right here,"},{"Start":"01:34.820 ","End":"01:39.950","Text":"that equals to 0.7 this equal to 0.7."},{"Start":"01:39.950 ","End":"01:43.010","Text":"Here, that\u0027s 0.3, that\u0027s 0.4,"},{"Start":"01:43.010 ","End":"01:45.575","Text":"and here that\u0027ll be 0.3."},{"Start":"01:45.575 ","End":"01:49.835","Text":"Now we\u0027re asked for the probability of exactly 1 event occurring."},{"Start":"01:49.835 ","End":"01:55.850","Text":"Well, 1 event occurring means that A occurs and not B that\u0027s this probability right here."},{"Start":"01:55.850 ","End":"02:01.565","Text":"Or that B occurs and A does not means that we\u0027re right here."},{"Start":"02:01.565 ","End":"02:07.535","Text":"Again, that\u0027s 0.3 plus 0.3 and that equals to 0.6."},{"Start":"02:07.535 ","End":"02:09.950","Text":"Now, we have to pay attention here."},{"Start":"02:09.950 ","End":"02:16.600","Text":"We\u0027re told that the events are mutually exclusive and not independent."},{"Start":"02:16.600 ","End":"02:20.850","Text":"That\u0027s the difference in the calculations here."}],"ID":13260},{"Watched":false,"Name":"Exercise 2","Duration":"3m 16s","ChapterTopicVideoID":12782,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"This statement asserts that in throwing a balanced die 4 times,"},{"Start":"00:03.330 ","End":"00:08.490","Text":"the probability that at least 2 results are the same is this ratio right here."},{"Start":"00:08.490 ","End":"00:11.220","Text":"Well, this statement is true."},{"Start":"00:11.220 ","End":"00:14.850","Text":"Now, why is that? Well, let\u0027s use our combinatorics here."},{"Start":"00:14.850 ","End":"00:18.090","Text":"The first thing that we want to do is we want to"},{"Start":"00:18.090 ","End":"00:22.520","Text":"know how many sets of 4s can I possibly have,"},{"Start":"00:22.520 ","End":"00:24.380","Text":"then that we\u0027re looking for the number of"},{"Start":"00:24.380 ","End":"00:28.175","Text":"possible results of throwing a balanced die times."},{"Start":"00:28.175 ","End":"00:31.845","Text":"Now, this will be our n of Omega,"},{"Start":"00:31.845 ","End":"00:34.700","Text":"that\u0027s the size of our sample space."},{"Start":"00:34.700 ","End":"00:40.250","Text":"Now, that equals to 6 times 6 times 6 times 6, and why is that?"},{"Start":"00:40.250 ","End":"00:42.425","Text":"Well, when you throw a balanced die once,"},{"Start":"00:42.425 ","End":"00:44.299","Text":"you have 6 possible results."},{"Start":"00:44.299 ","End":"00:46.325","Text":"So by throwing it 4 times,"},{"Start":"00:46.325 ","End":"00:49.080","Text":"you only have to multiply 6,"},{"Start":"00:49.080 ","End":"00:54.285","Text":"4 times, and that equals to 1,296."},{"Start":"00:54.285 ","End":"01:00.205","Text":"We\u0027re looking for the probability that at least 2 results are the same."},{"Start":"01:00.205 ","End":"01:03.570","Text":"This is what we\u0027re looking for."},{"Start":"01:03.570 ","End":"01:06.425","Text":"Now, let\u0027s have an example"},{"Start":"01:06.425 ","End":"01:10.280","Text":"of what does it mean when I say at least 2 results are the same."},{"Start":"01:10.280 ","End":"01:14.090","Text":"When I have 1 set of 4 where at least 2 results are the same,"},{"Start":"01:14.090 ","End":"01:16.760","Text":"it could be something like 3, 4, 3,"},{"Start":"01:16.760 ","End":"01:23.409","Text":"4 or another set of 4 where at least 2 results are the same would be 5,"},{"Start":"01:23.409 ","End":"01:25.125","Text":"5, 5, 5."},{"Start":"01:25.125 ","End":"01:31.235","Text":"These are examples of sets of 4s that I want to count in this probability."},{"Start":"01:31.235 ","End":"01:40.285","Text":"Now, that equals to 1 minus the probability that all results are different."},{"Start":"01:40.285 ","End":"01:43.550","Text":"Now what does that mean? Well, let\u0027s get an example of"},{"Start":"01:43.550 ","End":"01:46.040","Text":"a set of 4 where all the results are different."},{"Start":"01:46.040 ","End":"01:49.265","Text":"Well, it could be something like 1, 2, 3,"},{"Start":"01:49.265 ","End":"01:54.510","Text":"4, or 4, 5, 6, 1."},{"Start":"01:54.510 ","End":"02:01.310","Text":"These are results, 2 sets of 4s where the results are all different right here."},{"Start":"02:01.310 ","End":"02:08.900","Text":"Now, that equals to 1 minus something over 1,296."},{"Start":"02:08.900 ","End":"02:16.290","Text":"We\u0027re looking for how many sets of 4s do I have where all the results are different?"},{"Start":"02:16.460 ","End":"02:19.385","Text":"Now, what does that equal to?"},{"Start":"02:19.385 ","End":"02:21.604","Text":"Well, on the 1st throw,"},{"Start":"02:21.604 ","End":"02:24.020","Text":"I can have 6 possible results,"},{"Start":"02:24.020 ","End":"02:26.000","Text":"but in the 2nd throw,"},{"Start":"02:26.000 ","End":"02:29.040","Text":"I need 5 possible results."},{"Start":"02:29.040 ","End":"02:30.410","Text":"In the 3rd throw,"},{"Start":"02:30.410 ","End":"02:32.810","Text":"I have to have 4 possible results,"},{"Start":"02:32.810 ","End":"02:34.025","Text":"and in the 4th throw,"},{"Start":"02:34.025 ","End":"02:37.505","Text":"I have 3 possible results to choose from."},{"Start":"02:37.505 ","End":"02:43.740","Text":"All this comes out to 360."},{"Start":"02:44.300 ","End":"02:46.770","Text":"In this ratio right here,"},{"Start":"02:46.770 ","End":"02:51.750","Text":"that\u0027ll be 360 over 1,296,"},{"Start":"02:51.750 ","End":"02:59.465","Text":"that\u0027s the probability of getting a set of 4 where all results are different."},{"Start":"02:59.465 ","End":"03:01.725","Text":"Now, 1 minus this,"},{"Start":"03:01.725 ","End":"03:08.768","Text":"well that equals to 936 divided by 1,296."},{"Start":"03:08.768 ","End":"03:13.295","Text":"As we see, this ratio right here is the same as this one,"},{"Start":"03:13.295 ","End":"03:16.810","Text":"which makes this statement true."}],"ID":13261},{"Watched":false,"Name":"Exercise 3","Duration":"2m 26s","ChapterTopicVideoID":12783,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.543","Text":"Events A and B are independent events,"},{"Start":"00:02.543 ","End":"00:06.945","Text":"and their probabilities are 0.5 and 0.3, respectively."},{"Start":"00:06.945 ","End":"00:13.425","Text":"The statement asserts that the probability of at least one of them occurring is 0.8."},{"Start":"00:13.425 ","End":"00:18.465","Text":"Well, this statement is not true and why is that?"},{"Start":"00:18.465 ","End":"00:19.950","Text":"Well, let\u0027s write things down."},{"Start":"00:19.950 ","End":"00:28.485","Text":"The probability of A that equals to 0.5 and the probability of B that equals to 0.3."},{"Start":"00:28.485 ","End":"00:31.110","Text":"That\u0027s what we\u0027re given here and here."},{"Start":"00:31.110 ","End":"00:37.785","Text":"Now we\u0027re told that A and B are independent."},{"Start":"00:37.785 ","End":"00:41.055","Text":"That\u0027s what we\u0027re given. Now,"},{"Start":"00:41.055 ","End":"00:46.535","Text":"we\u0027re asked for the probability of at least one of them occurring."},{"Start":"00:46.535 ","End":"00:52.475","Text":"That means that we\u0027re looking for the probability of A union B."},{"Start":"00:52.475 ","End":"00:56.555","Text":"Now, according to the equation of union,"},{"Start":"00:56.555 ","End":"01:01.070","Text":"that\u0027ll be the probability of A plus the probability"},{"Start":"01:01.070 ","End":"01:06.125","Text":"of B minus the probability of A intersect B."},{"Start":"01:06.125 ","End":"01:09.440","Text":"Well, what\u0027s the probability of A intersect B?"},{"Start":"01:09.440 ","End":"01:12.515","Text":"Now, because A and B are independent,"},{"Start":"01:12.515 ","End":"01:15.860","Text":"then the probability of A intersect B,"},{"Start":"01:15.860 ","End":"01:21.170","Text":"that\u0027ll be equal to the probability of A times the probability of B"},{"Start":"01:21.170 ","End":"01:29.165","Text":"so that\u0027ll be equal to 0.5 times 0.3 and that equals to 0.15,"},{"Start":"01:29.165 ","End":"01:32.635","Text":"that\u0027s the probability of the intersect."},{"Start":"01:32.635 ","End":"01:34.880","Text":"Let\u0027s get back to here."},{"Start":"01:34.880 ","End":"01:39.245","Text":"Now this equals to the probability of A that\u0027s 0.5"},{"Start":"01:39.245 ","End":"01:44.930","Text":"plus the probability of B that\u0027s 0.3 minus 0.15,"},{"Start":"01:44.930 ","End":"01:47.285","Text":"the probability of A intersect B,"},{"Start":"01:47.285 ","End":"01:55.400","Text":"and that equals to 0.65 and this obviously does not equal to 0.8."},{"Start":"01:55.400 ","End":"01:59.645","Text":"That means that this statement is not true."},{"Start":"01:59.645 ","End":"02:09.775","Text":"Now, it would be true if we were told that the events A and B are mutually exclusive."},{"Start":"02:09.775 ","End":"02:11.405","Text":"But we weren\u0027t told that."},{"Start":"02:11.405 ","End":"02:16.190","Text":"We were just told that they were independent and that\u0027s the difference between"},{"Start":"02:16.190 ","End":"02:23.090","Text":"the calculations of probabilities of events that are mutually exclusive and independent."},{"Start":"02:23.090 ","End":"02:26.820","Text":"Here we\u0027re talking about independent events."}],"ID":13262},{"Watched":false,"Name":"Exercise 4","Duration":"4m 41s","ChapterTopicVideoID":12784,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.835","Text":"A and B are events in the sample space Omega."},{"Start":"00:02.835 ","End":"00:04.800","Text":"It\u0027s known that the probability of"},{"Start":"00:04.800 ","End":"00:09.210","Text":"A equals to the probability of B and that equals to 0.2."},{"Start":"00:09.210 ","End":"00:13.950","Text":"The statement asserts that if A and B are independent events,"},{"Start":"00:13.950 ","End":"00:19.020","Text":"then the probability of exactly 1 of them occurring is 0.4."},{"Start":"00:19.020 ","End":"00:23.010","Text":"Well, this statement is not true."},{"Start":"00:23.010 ","End":"00:25.620","Text":"Why is that? Well, let\u0027s write things down."},{"Start":"00:25.620 ","End":"00:27.930","Text":"The probability of A,"},{"Start":"00:27.930 ","End":"00:34.470","Text":"that equals to 0.2 and that equals to the probability of B as well."},{"Start":"00:34.470 ","End":"00:40.670","Text":"Now we\u0027re given that A and B are independent events."},{"Start":"00:40.670 ","End":"00:46.580","Text":"Now we\u0027re looking for the probability of exactly 1 of the events occurring."},{"Start":"00:46.580 ","End":"00:52.150","Text":"Well, that means that we\u0027re looking for the events A"},{"Start":"00:52.150 ","End":"00:59.940","Text":"and not B or not A and B."},{"Start":"00:59.940 ","End":"01:01.430","Text":"What we\u0027re looking for?"},{"Start":"01:01.430 ","End":"01:04.250","Text":"This probability right here."},{"Start":"01:04.250 ","End":"01:09.125","Text":"Well, let\u0027s solve this first of all using a Venn diagram."},{"Start":"01:09.125 ","End":"01:11.150","Text":"Here\u0027s our Venn diagram."},{"Start":"01:11.150 ","End":"01:13.720","Text":"When we\u0027re looking at this event right here,"},{"Start":"01:13.720 ","End":"01:15.230","Text":"A and not B,"},{"Start":"01:15.230 ","End":"01:19.480","Text":"what we\u0027re looking at this part of the event A."},{"Start":"01:19.480 ","End":"01:22.610","Text":"When we\u0027re looking at not A and B,"},{"Start":"01:22.610 ","End":"01:26.315","Text":"we\u0027re looking at this part of event B."},{"Start":"01:26.315 ","End":"01:29.225","Text":"Now the probability of a,"},{"Start":"01:29.225 ","End":"01:31.025","Text":"that equals to 0.2,"},{"Start":"01:31.025 ","End":"01:32.510","Text":"and the probability of B,"},{"Start":"01:32.510 ","End":"01:34.730","Text":"that also equals to 0.2."},{"Start":"01:34.730 ","End":"01:37.595","Text":"Now what about this intersect right here?"},{"Start":"01:37.595 ","End":"01:42.160","Text":"Well, we\u0027re looking then at the probability of A intersect B."},{"Start":"01:42.160 ","End":"01:45.710","Text":"Well, we know that A and B are independent,"},{"Start":"01:45.710 ","End":"01:51.230","Text":"so that means that equals to the probability of A times the probability of B."},{"Start":"01:51.230 ","End":"01:55.010","Text":"Now that equals to 0.2 times 0.2,"},{"Start":"01:55.010 ","End":"01:58.025","Text":"that equals to 0.04."},{"Start":"01:58.025 ","End":"02:03.375","Text":"The probability of A and not B,"},{"Start":"02:03.375 ","End":"02:12.690","Text":"well that\u0027s 0.2 minus 0.04 plus the probability of B,"},{"Start":"02:12.690 ","End":"02:19.700","Text":"0.2 minus the intersect, 0.04."},{"Start":"02:19.700 ","End":"02:25.850","Text":"Now that equals to 0.16 plus 0.16,"},{"Start":"02:25.850 ","End":"02:29.105","Text":"and that equals to 0.32."},{"Start":"02:29.105 ","End":"02:35.670","Text":"Now, that does not equal to 0.4 as the statement asserts."},{"Start":"02:35.670 ","End":"02:37.655","Text":"Therefore, this is not true."},{"Start":"02:37.655 ","End":"02:40.850","Text":"Now, we can look at this in a different way."},{"Start":"02:40.850 ","End":"02:44.530","Text":"We can use the probability matrix."},{"Start":"02:44.530 ","End":"02:46.940","Text":"Here\u0027s the probability matrix."},{"Start":"02:46.940 ","End":"02:50.795","Text":"We know that the sum of other probabilities that has to equal to 1,"},{"Start":"02:50.795 ","End":"02:54.590","Text":"the probability of A that equals to 0.2,"},{"Start":"02:54.590 ","End":"03:00.405","Text":"and the probability of B that also equals to 0.2."},{"Start":"03:00.405 ","End":"03:03.993","Text":"Let\u0027s just complete the probabilities here."},{"Start":"03:03.993 ","End":"03:07.790","Text":"That\u0027ll be 0.8 and that\u0027ll be 0.8."},{"Start":"03:07.790 ","End":"03:12.020","Text":"We\u0027re told that the A and B are independent,"},{"Start":"03:12.020 ","End":"03:13.895","Text":"so the intersect, well,"},{"Start":"03:13.895 ","End":"03:17.220","Text":"when we\u0027ve calculated that, that\u0027s 0.04."},{"Start":"03:17.240 ","End":"03:20.615","Text":"This equals to 0.16."},{"Start":"03:20.615 ","End":"03:23.450","Text":"Here, that equals to 0.16."},{"Start":"03:23.450 ","End":"03:27.590","Text":"Here, that will be equal to 0.64."},{"Start":"03:27.590 ","End":"03:30.785","Text":"Now again, we\u0027re looking for A and not B."},{"Start":"03:30.785 ","End":"03:34.065","Text":"That\u0027s this probability right here,"},{"Start":"03:34.065 ","End":"03:37.440","Text":"and we\u0027re looking for B and not A,"},{"Start":"03:37.440 ","End":"03:39.630","Text":"that\u0027s this probability right here."},{"Start":"03:39.630 ","End":"03:43.435","Text":"Again, that\u0027s 0.16 plus 0.16,"},{"Start":"03:43.435 ","End":"03:46.010","Text":"and that equals to 0.32,"},{"Start":"03:46.010 ","End":"03:49.295","Text":"which is not equal to 0.4."},{"Start":"03:49.295 ","End":"03:53.030","Text":"Now, what could you\u0027ve gone wrong?"},{"Start":"03:53.030 ","End":"04:00.400","Text":"Well, if you would have thought that A and B are mutually exclusive,"},{"Start":"04:00.400 ","End":"04:08.280","Text":"here\u0027s what the Venn diagram for A and B would be if A and B would be mutually exclusive."},{"Start":"04:08.280 ","End":"04:14.915","Text":"The probability of A would be 0.2 and the probability of B would be also 0.2."},{"Start":"04:14.915 ","End":"04:18.530","Text":"The probability of exactly 1 of them occurring,"},{"Start":"04:18.530 ","End":"04:23.230","Text":"that\u0027ll be the probability of A plus the probability of B."},{"Start":"04:23.230 ","End":"04:28.805","Text":"That would be equal to 0.2 plus 0.2 and that would be equal to 0.4."},{"Start":"04:28.805 ","End":"04:31.590","Text":"But we weren\u0027t told that."},{"Start":"04:31.590 ","End":"04:33.350","Text":"We weren\u0027t told that at all."},{"Start":"04:33.350 ","End":"04:36.665","Text":"We were told that A and B are independent,"},{"Start":"04:36.665 ","End":"04:39.125","Text":"not mutually exclusive, and therefore,"},{"Start":"04:39.125 ","End":"04:42.030","Text":"this statement is not true."}],"ID":13263},{"Watched":false,"Name":"Exercise 5","Duration":"1m 56s","ChapterTopicVideoID":12785,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.025","Text":"A spinning top has 4 faces."},{"Start":"00:02.025 ","End":"00:04.560","Text":"The statement asserts that the probability of getting"},{"Start":"00:04.560 ","End":"00:08.970","Text":"the same result when spinning the top 3 times is 1 over 16."},{"Start":"00:08.970 ","End":"00:11.940","Text":"Well, this statement is true."},{"Start":"00:11.940 ","End":"00:16.215","Text":"Now why is that? Well, let\u0027s use combinatorics to prove the statement."},{"Start":"00:16.215 ","End":"00:21.180","Text":"Now the first thing that I want to do is to calculate the size of the sample space."},{"Start":"00:21.180 ","End":"00:23.940","Text":"That will be n of omega."},{"Start":"00:23.940 ","End":"00:27.945","Text":"Now that equals to 4 times 4 times 4."},{"Start":"00:27.945 ","End":"00:30.780","Text":"That equals to 4^3. Now why is that?"},{"Start":"00:30.780 ","End":"00:33.105","Text":"Well, when I spin the top once,"},{"Start":"00:33.105 ","End":"00:37.505","Text":"I have 1 of 4 possible results,"},{"Start":"00:37.505 ","End":"00:40.310","Text":"and the same for the second and third spins,"},{"Start":"00:40.310 ","End":"00:43.265","Text":"so that\u0027ll be 4^3,"},{"Start":"00:43.265 ","End":"00:47.255","Text":"sets of 3 numbers."},{"Start":"00:47.255 ","End":"00:49.580","Text":"That will be my sample space."},{"Start":"00:49.580 ","End":"00:51.755","Text":"Now, what about what we\u0027re asked for?"},{"Start":"00:51.755 ","End":"00:55.940","Text":"The probability of getting the same result when spinning the top 3 times."},{"Start":"00:55.940 ","End":"01:03.785","Text":"I want to know how many sets of 3s have the same results when I spin the top 3 times."},{"Start":"01:03.785 ","End":"01:05.930","Text":"This is what I\u0027m looking for."},{"Start":"01:05.930 ","End":"01:07.880","Text":"Now, that equals to the following."},{"Start":"01:07.880 ","End":"01:11.060","Text":"In the first spin, well,"},{"Start":"01:11.060 ","End":"01:15.270","Text":"I can get 1 of 4 for results,"},{"Start":"01:15.270 ","End":"01:17.120","Text":"but in the second spin,"},{"Start":"01:17.120 ","End":"01:21.580","Text":"I need the same value as I had in the first spin,"},{"Start":"01:21.580 ","End":"01:24.949","Text":"and the same for the third spin as well,"},{"Start":"01:24.949 ","End":"01:27.485","Text":"so that equals to 4."},{"Start":"01:27.485 ","End":"01:33.920","Text":"What\u0027s the probability of getting the same result 3 times?"},{"Start":"01:33.920 ","End":"01:38.610","Text":"Well, that\u0027ll be equal to 4."},{"Start":"01:38.610 ","End":"01:42.800","Text":"That\u0027s the number of times that have the same result 3 times,"},{"Start":"01:42.800 ","End":"01:47.010","Text":"divided by my sample space: 4^3."},{"Start":"01:47.010 ","End":"01:48.120","Text":"Now let\u0027s simplify that."},{"Start":"01:48.120 ","End":"01:52.340","Text":"That would be 1 over 4 squared and that equals to 1 over 16,"},{"Start":"01:52.340 ","End":"01:56.880","Text":"so we can see that this statement is true."}],"ID":13264},{"Watched":false,"Name":"Exercise 6","Duration":"36s","ChapterTopicVideoID":12786,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"This statement asserts that if the expectation of the sum of"},{"Start":"00:02.970 ","End":"00:07.140","Text":"2 random variables equals to the sum of the expectation,"},{"Start":"00:07.140 ","End":"00:11.624","Text":"then X and Y are independent random variables."},{"Start":"00:11.624 ","End":"00:15.330","Text":"Well, this statement is not true."},{"Start":"00:15.330 ","End":"00:21.795","Text":"Why is that? This expression is true always."},{"Start":"00:21.795 ","End":"00:24.075","Text":"Now, as such,"},{"Start":"00:24.075 ","End":"00:29.385","Text":"this expression is not only true when X and Y are independent."},{"Start":"00:29.385 ","End":"00:32.520","Text":"It\u0027s true when they\u0027re dependent variables as well."},{"Start":"00:32.520 ","End":"00:36.730","Text":"Therefore, this statement right here is not true."}],"ID":13265},{"Watched":false,"Name":"Exercise 7","Duration":"1m 2s","ChapterTopicVideoID":12787,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.650","Text":"This statement asserts that the number of"},{"Start":"00:01.650 ","End":"00:04.980","Text":"different ways of putting 3 people in order is 9."},{"Start":"00:04.980 ","End":"00:09.210","Text":"Well, this statement is just not true. Why is that?"},{"Start":"00:09.210 ","End":"00:11.640","Text":"Well, let\u0027s use our combinatorics."},{"Start":"00:11.640 ","End":"00:17.535","Text":"The first person has 3 places in which he can be placed."},{"Start":"00:17.535 ","End":"00:22.500","Text":"The second person only has 2 places in which she can be placed,"},{"Start":"00:22.500 ","End":"00:26.100","Text":"and the third one doesn\u0027t have any choice whatsoever."},{"Start":"00:26.100 ","End":"00:29.370","Text":"We have here 3 times 2 times 1,"},{"Start":"00:29.370 ","End":"00:32.775","Text":"combinations of putting 3 people in order."},{"Start":"00:32.775 ","End":"00:35.510","Text":"That\u0027s 6 and that\u0027s not 9,"},{"Start":"00:35.510 ","End":"00:37.515","Text":"so therefore, this statement,"},{"Start":"00:37.515 ","End":"00:39.220","Text":"as we said, is not true."},{"Start":"00:39.220 ","End":"00:44.315","Text":"Now, in general, if we have n people or n things to be put in order,"},{"Start":"00:44.315 ","End":"00:52.960","Text":"well then there are n factorial number of combinations of putting these things in order."},{"Start":"00:52.960 ","End":"00:56.120","Text":"In our case, that would be 3 factorial,"},{"Start":"00:56.120 ","End":"00:58.835","Text":"which equals to 3 times 2 times 1."},{"Start":"00:58.835 ","End":"01:02.670","Text":"Again, that equals to 6, not 9."}],"ID":13266},{"Watched":false,"Name":"Exercise 8","Duration":"47s","ChapterTopicVideoID":12788,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.990","Text":"6 different toys must be divided amongst 4 girls and 2 boys."},{"Start":"00:03.990 ","End":"00:08.595","Text":"The statement asserts that the number of boys to divide the toys is 48."},{"Start":"00:08.595 ","End":"00:11.760","Text":"Well, this statement is not true."},{"Start":"00:11.760 ","End":"00:13.155","Text":"Now, why is that?"},{"Start":"00:13.155 ","End":"00:14.580","Text":"Well, here in this case,"},{"Start":"00:14.580 ","End":"00:16.875","Text":"it really doesn\u0027t matter if you\u0027re a boy or a girl."},{"Start":"00:16.875 ","End":"00:21.750","Text":"You want to distribute 6 different toys to 6 different people,"},{"Start":"00:21.750 ","End":"00:26.010","Text":"so the total number of ways to do so is 6 factorials,"},{"Start":"00:26.010 ","End":"00:30.195","Text":"just like taking the 6 toys and placing them in order."},{"Start":"00:30.195 ","End":"00:34.605","Text":"So 6 factorial, that equals to 6 times 5,"},{"Start":"00:34.605 ","End":"00:36.270","Text":"times 4, times 3,"},{"Start":"00:36.270 ","End":"00:38.160","Text":"times 2, times 1,"},{"Start":"00:38.160 ","End":"00:43.295","Text":"and that equals to 720 and not 48."},{"Start":"00:43.295 ","End":"00:47.610","Text":"Therefore, this statement is not true."}],"ID":13267},{"Watched":false,"Name":"Exercise 9","Duration":"1m 55s","ChapterTopicVideoID":12789,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.975","Text":"An ATM code is composed of 4 digits taken from 0-9."},{"Start":"00:03.975 ","End":"00:10.680","Text":"The statement asserts that the probability that all 4 numbers are different is 0.504."},{"Start":"00:10.680 ","End":"00:12.790","Text":"Well, this statement is true,"},{"Start":"00:12.790 ","End":"00:13.875","Text":"and why is that?"},{"Start":"00:13.875 ","End":"00:18.910","Text":"Well, the first thing that we need to do is to calculate our sample size."},{"Start":"00:19.250 ","End":"00:26.100","Text":"That\u0027ll be 10 times 10 times 10 times 10. Why is that?"},{"Start":"00:26.100 ","End":"00:28.650","Text":"Well, in the first place in the code, well,"},{"Start":"00:28.650 ","End":"00:32.025","Text":"we can have 10 digits, from 0-9."},{"Start":"00:32.025 ","End":"00:34.860","Text":"0 can be the first number in a code,"},{"Start":"00:34.860 ","End":"00:36.840","Text":"and what about the second place in the code?"},{"Start":"00:36.840 ","End":"00:40.880","Text":"Well, that also can have 10 digits and the third place and the fourth place."},{"Start":"00:40.880 ","End":"00:43.025","Text":"Now that equals to 10 to the fourth,"},{"Start":"00:43.025 ","End":"00:49.925","Text":"and that equals to 10,000 possible codes."},{"Start":"00:49.925 ","End":"00:56.230","Text":"Now, what about the event that all 4 numbers are different in the code?"},{"Start":"00:56.230 ","End":"01:01.110","Text":"Well, let\u0027s define A as the event that all digits are different."},{"Start":"01:01.110 ","End":"01:04.080","Text":"We\u0027re looking for n of A."},{"Start":"01:04.080 ","End":"01:06.110","Text":"The size of this event."},{"Start":"01:06.110 ","End":"01:09.530","Text":"Well, if all 4 digits are different,"},{"Start":"01:09.530 ","End":"01:12.335","Text":"then in the first place, right of the code,"},{"Start":"01:12.335 ","End":"01:16.430","Text":"then the first place can have 10 digits from 0-9."},{"Start":"01:16.430 ","End":"01:20.515","Text":"But the second place in the code can only have 9 digits,"},{"Start":"01:20.515 ","End":"01:23.340","Text":"and the third place can have 8 digits,"},{"Start":"01:23.340 ","End":"01:24.585","Text":"and the fourth place,"},{"Start":"01:24.585 ","End":"01:26.295","Text":"will have 7 digits."},{"Start":"01:26.295 ","End":"01:30.460","Text":"Now, that equals to 5,040."},{"Start":"01:30.590 ","End":"01:34.190","Text":"Now we\u0027re looking for the probability of A,"},{"Start":"01:34.190 ","End":"01:37.190","Text":"the probability that all digits are different, well,"},{"Start":"01:37.190 ","End":"01:42.845","Text":"that equals to 5,040 divided by this total sample size,"},{"Start":"01:42.845 ","End":"01:45.400","Text":"and that equals to 10,000,"},{"Start":"01:45.400 ","End":"01:50.915","Text":"and that means that this ratio equals to 0.504,"},{"Start":"01:50.915 ","End":"01:56.040","Text":"and that proves that this statement is true."}],"ID":13268},{"Watched":false,"Name":"Exercise 10","Duration":"3m 2s","ChapterTopicVideoID":12790,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.359","Text":"John and Mary go to the mall where there are a number of different activities."},{"Start":"00:03.359 ","End":"00:06.510","Text":"Now, we\u0027re given that the probability that they go bowling is"},{"Start":"00:06.510 ","End":"00:11.280","Text":"0.3 and the probability that they go out for coffee is 0.5,"},{"Start":"00:11.280 ","End":"00:14.790","Text":"and the probability that the go for at least 1 event,"},{"Start":"00:14.790 ","End":"00:17.160","Text":"bowling or coffee, is 0.7."},{"Start":"00:17.160 ","End":"00:23.815","Text":"Now, the assertion is that the probability that they go only for bowling is 0.3."},{"Start":"00:23.815 ","End":"00:25.940","Text":"Is that true or not?"},{"Start":"00:25.940 ","End":"00:29.455","Text":"Well, this is not true."},{"Start":"00:29.455 ","End":"00:32.510","Text":"Why is that? Well, let\u0027s see what\u0027s going on here."},{"Start":"00:32.510 ","End":"00:35.105","Text":"First of all, let\u0027s define our events."},{"Start":"00:35.105 ","End":"00:39.275","Text":"A would be going bowling,"},{"Start":"00:39.275 ","End":"00:44.530","Text":"and B, we\u0027ll define that as going out for coffee."},{"Start":"00:44.530 ","End":"00:47.115","Text":"Now, the probability of A,"},{"Start":"00:47.115 ","End":"00:49.275","Text":"that equals to 0.3,"},{"Start":"00:49.275 ","End":"00:50.580","Text":"that\u0027s given right here,"},{"Start":"00:50.580 ","End":"00:52.475","Text":"and the probability of B,"},{"Start":"00:52.475 ","End":"00:54.701","Text":"going out for coffee, that equals to 0.5,"},{"Start":"00:54.701 ","End":"00:59.060","Text":"and we\u0027re given that the probability of A union B,"},{"Start":"00:59.060 ","End":"01:01.670","Text":"that they go for at least 1 event,"},{"Start":"01:01.670 ","End":"01:02.960","Text":"either bowling or coffee, well,"},{"Start":"01:02.960 ","End":"01:06.450","Text":"that equals to 0.7."},{"Start":"01:06.450 ","End":"01:10.990","Text":"Now, there are many ways to solve this problem."},{"Start":"01:10.990 ","End":"01:14.910","Text":"Let\u0027s use the probability matrix."},{"Start":"01:14.910 ","End":"01:18.010","Text":"Here\u0027s a probability matrix and we know"},{"Start":"01:18.010 ","End":"01:20.920","Text":"that the probability sum up to 1. Now, what are we given?"},{"Start":"01:20.920 ","End":"01:24.355","Text":"That the probability of A is 0.3, that\u0027s right here,"},{"Start":"01:24.355 ","End":"01:29.260","Text":"and the probability of B that\u0027s 0.5, that\u0027s right here."},{"Start":"01:29.260 ","End":"01:31.840","Text":"Now, what else do we need to know?"},{"Start":"01:31.840 ","End":"01:35.315","Text":"Well, the probability of the union, that\u0027s 0.7."},{"Start":"01:35.315 ","End":"01:38.650","Text":"In order to calculate, for example,"},{"Start":"01:38.650 ","End":"01:41.980","Text":"the probability of not A and not B, well,"},{"Start":"01:41.980 ","End":"01:46.375","Text":"that\u0027s the complimentary set of A union B."},{"Start":"01:46.375 ","End":"01:54.170","Text":"What\u0027s the probability of not A and not B?"},{"Start":"01:54.170 ","End":"01:57.815","Text":"Well, as we said, that\u0027s the complimentary event to A union B."},{"Start":"01:57.815 ","End":"02:01.760","Text":"That\u0027ll be 1 minus the probability of A union B,"},{"Start":"02:01.760 ","End":"02:07.130","Text":"and that equals to 1 minus 0.7 and that equals to 0.3."},{"Start":"02:07.130 ","End":"02:11.515","Text":"This 0.3 will be right here, 0.3."},{"Start":"02:11.515 ","End":"02:14.450","Text":"Now, we can complete the matrix here."},{"Start":"02:14.450 ","End":"02:16.445","Text":"That\u0027ll be 0.7."},{"Start":"02:16.445 ","End":"02:20.630","Text":"Here, that\u0027ll be 0.4, 0.5."},{"Start":"02:20.630 ","End":"02:25.330","Text":"Here, that\u0027ll be 0.1, and here, that\u0027ll be 0.2."},{"Start":"02:25.330 ","End":"02:30.860","Text":"Now, we\u0027re asked about the probability that they go only bowling."},{"Start":"02:30.860 ","End":"02:33.620","Text":"That means that we\u0027re looking for the probability of A,"},{"Start":"02:33.620 ","End":"02:38.090","Text":"going bowling, and that they don\u0027t go out for coffee."},{"Start":"02:38.090 ","End":"02:41.435","Text":"We\u0027re looking at the event A and not B,"},{"Start":"02:41.435 ","End":"02:44.555","Text":"and we want to know what\u0027s that probability?"},{"Start":"02:44.555 ","End":"02:47.210","Text":"Well, A and not B, well,"},{"Start":"02:47.210 ","End":"02:52.650","Text":"that\u0027s right here, and that equals to 0.2."},{"Start":"02:52.650 ","End":"02:58.980","Text":"0.2 obviously doesn\u0027t equal to 0.3."},{"Start":"02:58.980 ","End":"03:03.630","Text":"That means that this assertion is not true."}],"ID":13269},{"Watched":false,"Name":"Exercise 11","Duration":"3m 4s","ChapterTopicVideoID":12791,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.695","Text":"There are 3 students in a class."},{"Start":"00:01.695 ","End":"00:05.580","Text":"The probability of a student passing an exam is 0.8."},{"Start":"00:05.580 ","End":"00:09.960","Text":"All the students are independent of each other and the statement asserts that"},{"Start":"00:09.960 ","End":"00:16.125","Text":"the probability of at least 1 student passing an exam is 0.992."},{"Start":"00:16.125 ","End":"00:20.285","Text":"Well, this statement is true and why is that?"},{"Start":"00:20.285 ","End":"00:23.480","Text":"Well, here we have a binomial distribution."},{"Start":"00:23.480 ","End":"00:25.805","Text":"Now, why is it a binomial distribution?"},{"Start":"00:25.805 ","End":"00:31.340","Text":"Because we have 3 independent Bernoulli trials."},{"Start":"00:31.340 ","End":"00:37.685","Text":"The trial here is whether you pass or fail the exam and we have 3 of them,"},{"Start":"00:37.685 ","End":"00:39.680","Text":"so n here that equals to 3,"},{"Start":"00:39.680 ","End":"00:43.605","Text":"the probability of success that\u0027s 0.8."},{"Start":"00:43.605 ","End":"00:46.205","Text":"Now we\u0027re interested in x,"},{"Start":"00:46.205 ","End":"00:50.965","Text":"and we\u0027ll define that as the number of students passing the exam."},{"Start":"00:50.965 ","End":"00:53.135","Text":"If we define it like this,"},{"Start":"00:53.135 ","End":"00:58.025","Text":"then the probability of x being less than or equal to k,"},{"Start":"00:58.025 ","End":"01:00.700","Text":"that equals to n over k,"},{"Start":"01:00.700 ","End":"01:05.740","Text":"p to the power of k times 1 minus p to the power of n minus"},{"Start":"01:05.740 ","End":"01:13.480","Text":"k. Now we\u0027re interested in the probability that at least 1 student passes."},{"Start":"01:13.480 ","End":"01:16.855","Text":"That means that x is greater or equal to 1."},{"Start":"01:16.855 ","End":"01:24.620","Text":"Now, that equals to 1 minus the probability of x being equal to 0."},{"Start":"01:24.620 ","End":"01:27.130","Text":"Let\u0027s just plug in the numbers here."},{"Start":"01:27.130 ","End":"01:29.230","Text":"That\u0027ll be 1 minus right here."},{"Start":"01:29.230 ","End":"01:32.350","Text":"Now, what\u0027s the probability of x equaling 0?"},{"Start":"01:32.350 ","End":"01:34.840","Text":"Well, that\u0027s this thing right here."},{"Start":"01:34.840 ","End":"01:37.100","Text":"That\u0027ll be 3 over 0,"},{"Start":"01:37.100 ","End":"01:44.300","Text":"0.80 to the power of 0 times 0.2 to the power of 3."},{"Start":"01:44.300 ","End":"01:46.325","Text":"Now 3 over 0, that\u0027s 1."},{"Start":"01:46.325 ","End":"01:49.070","Text":"0.8 to the power of 0, that\u0027s also 1."},{"Start":"01:49.070 ","End":"01:53.940","Text":"So 1 times 1 times 0.2 to the power of 3,"},{"Start":"01:53.940 ","End":"01:59.090","Text":"that\u0027ll be 1 minus then 0.2 to the power of 3,"},{"Start":"01:59.090 ","End":"02:04.070","Text":"and that equals to 0.992."},{"Start":"02:04.070 ","End":"02:05.480","Text":"Here we go."},{"Start":"02:05.480 ","End":"02:07.010","Text":"This assertion is true."},{"Start":"02:07.010 ","End":"02:10.820","Text":"Now, we could have solved this in a different way."},{"Start":"02:10.820 ","End":"02:16.355","Text":"All I have to do is calculate the probability of failure."},{"Start":"02:16.355 ","End":"02:18.530","Text":"What\u0027s the probability of failure?"},{"Start":"02:18.530 ","End":"02:26.104","Text":"That\u0027s 0.2, that\u0027s the probability of fail."},{"Start":"02:26.104 ","End":"02:31.712","Text":"Now, what\u0027s the probability of failure for all 3 students?"},{"Start":"02:31.712 ","End":"02:35.540","Text":"Well, that\u0027ll be equal to 0.2 to the power of 3."},{"Start":"02:35.540 ","End":"02:36.995","Text":"The first student fails,"},{"Start":"02:36.995 ","End":"02:38.810","Text":"and then the second student fails,"},{"Start":"02:38.810 ","End":"02:45.935","Text":"and then the third student fails and now that equals to 0.008."},{"Start":"02:45.935 ","End":"02:48.990","Text":"Now, if at least 1 student passes,"},{"Start":"02:48.990 ","End":"02:54.570","Text":"that\u0027ll be 1 minus the probability that the student fails."},{"Start":"02:54.680 ","End":"03:00.930","Text":"That equals to 1 minus 0.2 to the power of 3 and again,"},{"Start":"03:00.930 ","End":"03:04.540","Text":"that equals to 0.992."}],"ID":13270},{"Watched":false,"Name":"Exercise 12","Duration":"59s","ChapterTopicVideoID":12792,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"A restaurant offers 3 first courses,"},{"Start":"00:02.310 ","End":"00:04.380","Text":"4 entrées, and 2 desserts."},{"Start":"00:04.380 ","End":"00:09.060","Text":"The assertion is, is that the total number of combinations of ordering"},{"Start":"00:09.060 ","End":"00:13.840","Text":"1 first course plus 1 entrée plus 1 dessert is 9."},{"Start":"00:13.840 ","End":"00:17.385","Text":"Well, this is not true."},{"Start":"00:17.385 ","End":"00:21.210","Text":"Now, this is a classical question in combinatorics."},{"Start":"00:21.210 ","End":"00:24.765","Text":"Now, how many options do we have to choose the first course?"},{"Start":"00:24.765 ","End":"00:29.025","Text":"Well, we have 3. In entrée, well, we have 4."},{"Start":"00:29.025 ","End":"00:30.975","Text":"In the dessert, we have 2."},{"Start":"00:30.975 ","End":"00:33.360","Text":"So 3 times 4, that\u0027s 12,"},{"Start":"00:33.360 ","End":"00:35.460","Text":"times 2, that\u0027s 24."},{"Start":"00:35.460 ","End":"00:40.950","Text":"We have 24 options of choosing any first course,"},{"Start":"00:40.950 ","End":"00:44.915","Text":"plus any entrée, plus any dessert."},{"Start":"00:44.915 ","End":"00:47.280","Text":"It doesn\u0027t equal to 9."},{"Start":"00:47.280 ","End":"00:49.864","Text":"This plus here is misleading."},{"Start":"00:49.864 ","End":"00:52.317","Text":"When we\u0027re dealing with combinatorics,"},{"Start":"00:52.317 ","End":"00:54.770","Text":"when we try to calculate combinations,"},{"Start":"00:54.770 ","End":"00:57.590","Text":"we multiply the combinations,"},{"Start":"00:57.590 ","End":"00:59.880","Text":"we don\u0027t add them."}],"ID":13271},{"Watched":false,"Name":"Exercise 13","Duration":"3m 2s","ChapterTopicVideoID":12793,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.260","Text":"In a shooting competition,"},{"Start":"00:01.260 ","End":"00:06.240","Text":"a competitor plays until he hits the target but doesn\u0027t shoot more than 4 times."},{"Start":"00:06.240 ","End":"00:10.200","Text":"The probability of a competitor hitting the target is 0.6,"},{"Start":"00:10.200 ","End":"00:13.950","Text":"and the assertion is that the chances that a competitor will shoot at"},{"Start":"00:13.950 ","End":"00:18.640","Text":"the target 4 times is 0.064."},{"Start":"00:18.640 ","End":"00:22.800","Text":"Let\u0027s use a probability tree to calculate what is"},{"Start":"00:22.800 ","End":"00:27.735","Text":"the probability of shooting at the target on the fourth try."},{"Start":"00:27.735 ","End":"00:30.240","Text":"Here\u0027s a probability tree,"},{"Start":"00:30.240 ","End":"00:32.645","Text":"and here we see the first try right here."},{"Start":"00:32.645 ","End":"00:38.915","Text":"He either succeeds with a probability of 0.6 or he fails with a probability of 0.4."},{"Start":"00:38.915 ","End":"00:41.105","Text":"Now if he succeeds, he stops,"},{"Start":"00:41.105 ","End":"00:42.560","Text":"and if he fails, well,"},{"Start":"00:42.560 ","End":"00:44.485","Text":"he moves on to the second try."},{"Start":"00:44.485 ","End":"00:49.294","Text":"Again here, he has the probability of succeeding or failing."},{"Start":"00:49.294 ","End":"00:51.150","Text":"If he succeeds, then he stops."},{"Start":"00:51.150 ","End":"00:54.015","Text":"If he fails, he goes out to the third try."},{"Start":"00:54.015 ","End":"00:59.630","Text":"Again, right here, you can either succeed or fail with a probability of 0.6 or 0.4."},{"Start":"00:59.630 ","End":"01:01.460","Text":"If he succeeds, then he stops."},{"Start":"01:01.460 ","End":"01:02.900","Text":"If he fails,"},{"Start":"01:02.900 ","End":"01:05.210","Text":"then he goes on to the fourth try."},{"Start":"01:05.210 ","End":"01:08.835","Text":"Here, he can either succeed or fail,"},{"Start":"01:08.835 ","End":"01:13.140","Text":"but in both cases, the competitor stops."},{"Start":"01:13.140 ","End":"01:21.305","Text":"In order to see what\u0027s the probability of shooting at the target on the fourth try,"},{"Start":"01:21.305 ","End":"01:24.665","Text":"well, then there are 2 paths on this tree."},{"Start":"01:24.665 ","End":"01:29.390","Text":"The first path is where he doesn\u0027t succeed on"},{"Start":"01:29.390 ","End":"01:35.350","Text":"the first 3 tries and in the fourth try he doesn\u0027t succeed,"},{"Start":"01:35.350 ","End":"01:41.910","Text":"or he doesn\u0027t succeed on the first 3 tries and on the fourth try he does succeed."},{"Start":"01:41.910 ","End":"01:47.675","Text":"Now, what we have to do is we have to calculate the probabilities of these 2 paths."},{"Start":"01:47.675 ","End":"01:49.235","Text":"Now, let\u0027s take the path,"},{"Start":"01:49.235 ","End":"01:52.295","Text":"the yellow path where he doesn\u0027t succeed and all 4 tries."},{"Start":"01:52.295 ","End":"01:57.630","Text":"That\u0027ll be 0.4 times 0.4 4 times."},{"Start":"01:57.630 ","End":"02:01.392","Text":"That means that\u0027s 0.4 to the power of 4."},{"Start":"02:01.392 ","End":"02:04.669","Text":"Plus, now let\u0027s take a look at this path right here,"},{"Start":"02:04.669 ","End":"02:08.235","Text":"he doesn\u0027t succeed 3 times."},{"Start":"02:08.235 ","End":"02:14.550","Text":"That\u0027s 0.4 to the power of 3 and he succeeds on the fourth try, so that\u0027s 0.6."},{"Start":"02:14.550 ","End":"02:16.850","Text":"Now, the probability,"},{"Start":"02:16.850 ","End":"02:18.230","Text":"this probability right here,"},{"Start":"02:18.230 ","End":"02:22.205","Text":"that equals to 0.064."},{"Start":"02:22.205 ","End":"02:25.475","Text":"So this assertion is true."},{"Start":"02:25.475 ","End":"02:28.700","Text":"Now, there\u0027s another way of looking at this."},{"Start":"02:28.700 ","End":"02:33.020","Text":"If he doesn\u0027t succeed 3 times in a row,"},{"Start":"02:33.020 ","End":"02:35.990","Text":"0.4 to the power of 3, well,"},{"Start":"02:35.990 ","End":"02:40.760","Text":"then it really doesn\u0027t matter whether he succeeds or fails on the fourth try."},{"Start":"02:40.760 ","End":"02:46.390","Text":"He got to the fourth try and he shoots for the fourth time."},{"Start":"02:46.390 ","End":"02:49.530","Text":"That means that this thing right here,"},{"Start":"02:49.530 ","End":"02:54.670","Text":"that also equals to 0.064."},{"Start":"02:55.370 ","End":"02:58.965","Text":"In both ways of looking at this,"},{"Start":"02:58.965 ","End":"03:02.560","Text":"we see that this assertion is true."}],"ID":13272},{"Watched":false,"Name":"Exercise 14","Duration":"4m 24s","ChapterTopicVideoID":12794,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.900","Text":"The probability that a student would study on a given day is 0.7"},{"Start":"00:03.900 ","End":"00:08.500","Text":"if his mother asks and 0.4 if his mother doesn\u0027t ask."},{"Start":"00:08.500 ","End":"00:12.615","Text":"The student\u0027s mom asks him to study on 60 percent of the days."},{"Start":"00:12.615 ","End":"00:16.515","Text":"Now, a friend came to visit the student and found that he was studying."},{"Start":"00:16.515 ","End":"00:19.440","Text":"The assertion is that the probability that"},{"Start":"00:19.440 ","End":"00:24.120","Text":"his mother asked him to study on that specific day is 0.742."},{"Start":"00:24.120 ","End":"00:27.240","Text":"Well, this assertion is true."},{"Start":"00:27.240 ","End":"00:29.820","Text":"Now, how do we know? How do we prove that?"},{"Start":"00:29.820 ","End":"00:33.750","Text":"Well, the first thing that we need to do is to determine the technique that we\u0027ll be"},{"Start":"00:33.750 ","End":"00:37.695","Text":"using in order to calculate this probability."},{"Start":"00:37.695 ","End":"00:43.100","Text":"Now, the best technique for this type of question is a probability tree."},{"Start":"00:43.100 ","End":"00:45.680","Text":"Why is that? Well, we have stages here."},{"Start":"00:45.680 ","End":"00:51.380","Text":"We can see that we have the first stage where mom asks and or doesn\u0027t ask."},{"Start":"00:51.380 ","End":"00:53.120","Text":"Then in each case,"},{"Start":"00:53.120 ","End":"00:55.265","Text":"whether the student studies or not."},{"Start":"00:55.265 ","End":"00:58.420","Text":"Let\u0027s build our probability tree."},{"Start":"00:58.420 ","End":"01:03.845","Text":"Here\u0027s our tree, and let\u0027s now fill it with our probabilities."},{"Start":"01:03.845 ","End":"01:05.645","Text":"Now, the student will study,"},{"Start":"01:05.645 ","End":"01:10.370","Text":"that the probability that the student was study is 0.7 if mom asks,"},{"Start":"01:10.370 ","End":"01:13.025","Text":"so here\u0027s the branch where mom asks,"},{"Start":"01:13.025 ","End":"01:15.050","Text":"and here\u0027s where the students study."},{"Start":"01:15.050 ","End":"01:18.430","Text":"Now the probability here would be 0.7."},{"Start":"01:18.430 ","End":"01:23.490","Text":"If mom asks and the student didn\u0027t study, well that\u0027ll be 0.3."},{"Start":"01:23.490 ","End":"01:29.260","Text":"Now, if mom didn\u0027t ask and the student actually did study,"},{"Start":"01:29.260 ","End":"01:31.540","Text":"the probability there would be 0.4,"},{"Start":"01:31.540 ","End":"01:33.385","Text":"so that\u0027s 0.4 here."},{"Start":"01:33.385 ","End":"01:35.530","Text":"Mom didn\u0027t ask and the student studied,"},{"Start":"01:35.530 ","End":"01:37.765","Text":"that\u0027s the probability of him studying."},{"Start":"01:37.765 ","End":"01:43.970","Text":"If mom doesn\u0027t ask and the student didn\u0027t study, that\u0027s 0.6."},{"Start":"01:43.970 ","End":"01:49.630","Text":"Now we\u0027re also given that the student\u0027s mom asks him to study on 60 percent of the days."},{"Start":"01:49.630 ","End":"01:50.845","Text":"So if mom asks,"},{"Start":"01:50.845 ","End":"01:54.040","Text":"that\u0027s the probability of mom asking, 60 percent."},{"Start":"01:54.040 ","End":"01:59.375","Text":"The probability of mom not asking is 0.4."},{"Start":"01:59.375 ","End":"02:01.860","Text":"Now we have all the probabilities."},{"Start":"02:01.860 ","End":"02:05.865","Text":"The next step is to see what we\u0027re looking for."},{"Start":"02:05.865 ","End":"02:12.155","Text":"Well, we\u0027re looking for the probability that the mother asks him to study,"},{"Start":"02:12.155 ","End":"02:16.905","Text":"given that the student actually studied."},{"Start":"02:16.905 ","End":"02:20.645","Text":"A friend came to visit this student and found that he was studying."},{"Start":"02:20.645 ","End":"02:22.610","Text":"Given that the student was studying,"},{"Start":"02:22.610 ","End":"02:25.865","Text":"what\u0027s the probability that the mother asked him to study?"},{"Start":"02:25.865 ","End":"02:27.170","Text":"That\u0027s what we\u0027re looking for."},{"Start":"02:27.170 ","End":"02:30.595","Text":"This is a conditional probability."},{"Start":"02:30.595 ","End":"02:34.010","Text":"We\u0027re looking for the probability of mom"},{"Start":"02:34.010 ","End":"02:39.300","Text":"asking given that the student was actually studying."},{"Start":"02:39.300 ","End":"02:43.455","Text":"Now, this is something that we know how to calculate."},{"Start":"02:43.455 ","End":"02:49.805","Text":"In the denominator, we\u0027ll have the probability of studying."},{"Start":"02:49.805 ","End":"02:52.145","Text":"In the numerator, well,"},{"Start":"02:52.145 ","End":"02:53.750","Text":"we\u0027ll have the intersect,"},{"Start":"02:53.750 ","End":"03:00.875","Text":"so that\u0027ll be the probability of mom asking intersect the student studying."},{"Start":"03:00.875 ","End":"03:07.080","Text":"Now, we can calculate these probabilities by using this tree. Let\u0027s do that."},{"Start":"03:08.540 ","End":"03:13.475","Text":"What\u0027s the probability of the student studying?"},{"Start":"03:13.475 ","End":"03:20.060","Text":"Well here, we have to look at this thing right here and this thing right here."},{"Start":"03:20.060 ","End":"03:25.639","Text":"What are the paths leading to this branch right here?"},{"Start":"03:25.639 ","End":"03:30.380","Text":"Well, it\u0027s this path right here and this path right here."},{"Start":"03:30.380 ","End":"03:32.870","Text":"What are the probabilities here?"},{"Start":"03:32.870 ","End":"03:36.620","Text":"Well, that\u0027ll be 0.4 times"},{"Start":"03:36.620 ","End":"03:46.495","Text":"0.4 plus 0.6 times 0.7."},{"Start":"03:46.495 ","End":"03:48.540","Text":"That\u0027s in the denominator."},{"Start":"03:48.540 ","End":"03:50.195","Text":"Now what about the numerator?"},{"Start":"03:50.195 ","End":"03:54.835","Text":"Well, that\u0027s the probability of mom asking and him studying."},{"Start":"03:54.835 ","End":"03:58.745","Text":"That\u0027s mom asking and that\u0027s the student studying."},{"Start":"03:58.745 ","End":"04:00.380","Text":"That\u0027s this path right here."},{"Start":"04:00.380 ","End":"04:02.480","Text":"What\u0027s the probability there?"},{"Start":"04:02.480 ","End":"04:06.410","Text":"That\u0027s 0.6 times 0.7."},{"Start":"04:06.410 ","End":"04:08.960","Text":"Now, having calculated this,"},{"Start":"04:08.960 ","End":"04:13.865","Text":"that comes out to 0.742."},{"Start":"04:13.865 ","End":"04:18.470","Text":"This then is the probability that mom"},{"Start":"04:18.470 ","End":"04:24.600","Text":"asks him to study given that the student actually did study."}],"ID":13273},{"Watched":false,"Name":"Exercise 15","Duration":"1m 11s","ChapterTopicVideoID":12795,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.100","Text":"This statement asserts that the number of"},{"Start":"00:02.100 ","End":"00:04.710","Text":"initiatives that can be created in an alphabet having"},{"Start":"00:04.710 ","End":"00:10.035","Text":"26 letters for the first and last name is 52."},{"Start":"00:10.035 ","End":"00:14.250","Text":"This statement is not true."},{"Start":"00:14.250 ","End":"00:16.935","Text":"Why is that? Here,"},{"Start":"00:16.935 ","End":"00:18.390","Text":"we\u0027ll be using combinatorics."},{"Start":"00:18.390 ","End":"00:24.345","Text":"We\u0027re looking for the number of combinations of initials for the first and last names."},{"Start":"00:24.345 ","End":"00:26.805","Text":"Basically, it\u0027s a 2-step process."},{"Start":"00:26.805 ","End":"00:30.540","Text":"The first step, we\u0027re looking at the number of"},{"Start":"00:30.540 ","End":"00:36.995","Text":"letters that I can have as an initial for the first name, that\u0027ll be 26."},{"Start":"00:36.995 ","End":"00:40.920","Text":"In the next step, I\u0027ll be looking at the number of"},{"Start":"00:40.920 ","End":"00:44.480","Text":"initials that I can have in the last name."},{"Start":"00:44.480 ","End":"00:47.215","Text":"That\u0027s also 26."},{"Start":"00:47.215 ","End":"00:51.470","Text":"The number of combinations for initials for"},{"Start":"00:51.470 ","End":"00:55.400","Text":"the first and last name in an alphabet having 26 letters,"},{"Start":"00:55.400 ","End":"01:01.140","Text":"that\u0027ll be 26 times 26 and that"},{"Start":"01:01.140 ","End":"01:08.100","Text":"comes out to 676, not 52."},{"Start":"01:08.100 ","End":"01:12.660","Text":"So as we said, this statement is not true."}],"ID":13274},{"Watched":false,"Name":"Exercise 16","Duration":"1m 29s","ChapterTopicVideoID":12796,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"This statement asserts that the number of 3 digit numbers"},{"Start":"00:03.810 ","End":"00:08.760","Text":"in which the digits are all different from each other is 648."},{"Start":"00:08.760 ","End":"00:12.075","Text":"Okay, well, this statement is true."},{"Start":"00:12.075 ","End":"00:15.840","Text":"Now, why is that? Well, I\u0027ll be using some combinatorics here."},{"Start":"00:15.840 ","End":"00:19.445","Text":"The first thing that we need to do is to know"},{"Start":"00:19.445 ","End":"00:24.200","Text":"how many digits do I have in the hundreds place? We\u0027ll call that N1."},{"Start":"00:24.200 ","End":"00:27.570","Text":"Well, I have 10 digits from 0-9,"},{"Start":"00:27.570 ","End":"00:31.910","Text":"but I can only use 9 digits. Why is that?"},{"Start":"00:31.910 ","End":"00:35.885","Text":"Because I can\u0027t use the 0 number here in the hundreds place."},{"Start":"00:35.885 ","End":"00:38.315","Text":"What about the tens place?"},{"Start":"00:38.315 ","End":"00:41.735","Text":"Well, here, how many digits do I have?"},{"Start":"00:41.735 ","End":"00:44.390","Text":"Well, again, I have 10 digits here,"},{"Start":"00:44.390 ","End":"00:47.345","Text":"but it can only choose from 9 of them."},{"Start":"00:47.345 ","End":"00:48.613","Text":"Again, why is that?"},{"Start":"00:48.613 ","End":"00:51.684","Text":"Well, 0 can be used in the tens place,"},{"Start":"00:51.684 ","End":"00:56.210","Text":"but I can\u0027t use the digit that was already chosen in the hundreds place."},{"Start":"00:56.210 ","End":"01:00.710","Text":"Again, because we\u0027re looking for digits that are all different from each other."},{"Start":"01:00.710 ","End":"01:02.720","Text":"In the ones place,"},{"Start":"01:02.720 ","End":"01:05.540","Text":"well, how many digits do I have to choose from?"},{"Start":"01:05.540 ","End":"01:08.600","Text":"Well, I have 8 digits to choose from, right?"},{"Start":"01:08.600 ","End":"01:13.370","Text":"Not the digits from the hundreds place or the tens place."},{"Start":"01:13.370 ","End":"01:14.690","Text":"I have 10 digits,"},{"Start":"01:14.690 ","End":"01:17.495","Text":"so I have 8 digits to choose from."},{"Start":"01:17.495 ","End":"01:21.022","Text":"Again, how many combinations in total?"},{"Start":"01:21.022 ","End":"01:23.710","Text":"Well, that\u0027ll be 9 times 9 times 8,"},{"Start":"01:23.710 ","End":"01:27.170","Text":"and that equals to 648,"},{"Start":"01:27.170 ","End":"01:30.390","Text":"which makes this assertion true."}],"ID":13275},{"Watched":false,"Name":"Exercise 17","Duration":"1m 1s","ChapterTopicVideoID":12797,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.804","Text":"Events A and B are mutually exclusive."},{"Start":"00:02.804 ","End":"00:06.270","Text":"Now, the chances of each of them occurring are 0.3."},{"Start":"00:06.270 ","End":"00:11.835","Text":"The assertion is that the chances of at least 1 of them occurring are 0.6."},{"Start":"00:11.835 ","End":"00:15.840","Text":"We know that A and B are mutually exclusive and we\u0027re given that"},{"Start":"00:15.840 ","End":"00:21.610","Text":"the probability of A equals to the probability of B and that equals to 0.3."},{"Start":"00:21.610 ","End":"00:27.560","Text":"Now, we\u0027re asked about the probability of at least 1 of them occurring."},{"Start":"00:27.560 ","End":"00:30.395","Text":"That means that we\u0027re looking at A union B."},{"Start":"00:30.395 ","End":"00:35.690","Text":"Now, the equation for this is the probability of A plus"},{"Start":"00:35.690 ","End":"00:41.330","Text":"the probability of B minus the probability of A intersect B."},{"Start":"00:41.330 ","End":"00:44.075","Text":"Now, because A and B are mutually exclusive,"},{"Start":"00:44.075 ","End":"00:45.530","Text":"then the intersect,"},{"Start":"00:45.530 ","End":"00:47.330","Text":"that equals to 0."},{"Start":"00:47.330 ","End":"00:52.550","Text":"That means that we\u0027re looking at the probability of A plus the probability of B,"},{"Start":"00:52.550 ","End":"00:55.955","Text":"that\u0027s 0.3 plus 0.3,"},{"Start":"00:55.955 ","End":"00:58.190","Text":"and that equals to 0.6,"},{"Start":"00:58.190 ","End":"01:02.080","Text":"so this assertion is true."}],"ID":13276},{"Watched":false,"Name":"Exercise 18","Duration":"2m 7s","ChapterTopicVideoID":12798,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"A, B, and C are 3 events in the sample space Omega."},{"Start":"00:03.075 ","End":"00:06.420","Text":"Now it\u0027s known that the probability of A, B,"},{"Start":"00:06.420 ","End":"00:11.535","Text":"and C, they equal to each other and they are equal to 0.2."},{"Start":"00:11.535 ","End":"00:16.650","Text":"The probability of only event B occurring if the events are independent,"},{"Start":"00:16.650 ","End":"00:21.990","Text":"when the assertion is that probability is equal to 0.2."},{"Start":"00:21.990 ","End":"00:27.000","Text":"This is not true. Now, why is that?"},{"Start":"00:27.000 ","End":"00:32.400","Text":"Well, we\u0027re looking at the probability of B occurring,"},{"Start":"00:32.400 ","End":"00:38.465","Text":"but not A occurring and not C occurring,"},{"Start":"00:38.465 ","End":"00:42.770","Text":"that\u0027s only the probability of only B occurring."},{"Start":"00:42.770 ","End":"00:44.570","Text":"Now because A, B,"},{"Start":"00:44.570 ","End":"00:45.800","Text":"and C are independent,"},{"Start":"00:45.800 ","End":"00:48.170","Text":"then we can multiply the probabilities,"},{"Start":"00:48.170 ","End":"00:52.205","Text":"that\u0027ll be the probability of B times the probability"},{"Start":"00:52.205 ","End":"00:56.510","Text":"of not A times the probability of not C. Well,"},{"Start":"00:56.510 ","End":"00:58.840","Text":"what\u0027s the probability of not A?"},{"Start":"00:58.840 ","End":"01:01.365","Text":"The probability of not A,"},{"Start":"01:01.365 ","End":"01:04.895","Text":"that equals to 1 minus the probability of A."},{"Start":"01:04.895 ","End":"01:08.345","Text":"That equals to 1 minus 0.2."},{"Start":"01:08.345 ","End":"01:11.450","Text":"That\u0027s right here, and that equals to 0.8."},{"Start":"01:11.450 ","End":"01:13.940","Text":"The same thing for not C."},{"Start":"01:13.940 ","End":"01:18.140","Text":"The probability of not C that equals to 1 minus the probability of C,"},{"Start":"01:18.140 ","End":"01:22.025","Text":"that equals to 1 minus 0.2 and that equals to 0.8."},{"Start":"01:22.025 ","End":"01:23.855","Text":"Getting back to here,"},{"Start":"01:23.855 ","End":"01:25.970","Text":"the probability of B,"},{"Start":"01:25.970 ","End":"01:30.590","Text":"that equals to 0.2 times the probability of not A,"},{"Start":"01:30.590 ","End":"01:33.530","Text":"that\u0027s 0.8 times the probability of not C,"},{"Start":"01:33.530 ","End":"01:35.315","Text":"that\u0027s also 0.8,"},{"Start":"01:35.315 ","End":"01:39.210","Text":"and that does not equal to 0.2."},{"Start":"01:39.210 ","End":"01:41.945","Text":"Therefore, this assertion is not true."},{"Start":"01:41.945 ","End":"01:44.450","Text":"Now, where could you have gone wrong here?"},{"Start":"01:44.450 ","End":"01:48.780","Text":"Well, the probability of B occurring,"},{"Start":"01:48.780 ","End":"01:52.160","Text":"that does is equal to 0.2 that\u0027s given to us."},{"Start":"01:52.160 ","End":"01:53.705","Text":"But we weren\u0027t asked that."},{"Start":"01:53.705 ","End":"01:58.175","Text":"We were asked about the probability of only event B occurring."},{"Start":"01:58.175 ","End":"02:02.645","Text":"That means that we\u0027re looking for the probability of B and not"},{"Start":"02:02.645 ","End":"02:08.010","Text":"A and not C. This is the correct calculation."}],"ID":13277},{"Watched":false,"Name":"Exercise 19","Duration":"2m 57s","ChapterTopicVideoID":12799,"CourseChapterTopicPlaylistID":245057,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.260","Text":"In a certain population,"},{"Start":"00:01.260 ","End":"00:04.755","Text":"the distribution of people according to their blood type is as follows."},{"Start":"00:04.755 ","End":"00:05.985","Text":"Here we have a table,"},{"Start":"00:05.985 ","End":"00:08.970","Text":"in the first column that\u0027s the blood type A, O, B,"},{"Start":"00:08.970 ","End":"00:11.250","Text":"and AB, and in the second column,"},{"Start":"00:11.250 ","End":"00:13.740","Text":"we have the percentage of the population."},{"Start":"00:13.740 ","End":"00:17.580","Text":"40 percent of the population have blood type A,"},{"Start":"00:17.580 ","End":"00:19.725","Text":"30 percent have blood type O,"},{"Start":"00:19.725 ","End":"00:21.135","Text":"and so on and so forth."},{"Start":"00:21.135 ","End":"00:26.370","Text":"Now, 4 people are randomly selected from this population and the assertion"},{"Start":"00:26.370 ","End":"00:31.437","Text":"is that the probability that exactly one of them has blood type A,"},{"Start":"00:31.437 ","End":"00:34.115","Text":"well, that equals to 0.4."},{"Start":"00:34.115 ","End":"00:40.175","Text":"Well, this assertion is not true and why is that?"},{"Start":"00:40.175 ","End":"00:41.855","Text":"Well, first of all,"},{"Start":"00:41.855 ","End":"00:46.940","Text":"we have to identify that we\u0027re dealing with a binomial distribution. Why is that?"},{"Start":"00:46.940 ","End":"00:52.530","Text":"Well, we\u0027re looking at four Bernoulli trials."},{"Start":"00:52.660 ","End":"00:57.350","Text":"Now, the success of a Bernoulli trial is defined as having"},{"Start":"00:57.350 ","End":"01:01.490","Text":"blood type A and the probability of having blood type A,"},{"Start":"01:01.490 ","End":"01:05.270","Text":"well, that\u0027s given to us right here in the statement, that\u0027s 0.4."},{"Start":"01:05.270 ","End":"01:08.780","Text":"By the way, the failure, that\u0027s q,"},{"Start":"01:08.780 ","End":"01:11.135","Text":"that equals to 1 minus p,"},{"Start":"01:11.135 ","End":"01:13.580","Text":"that equals to 1 minus 0.4,"},{"Start":"01:13.580 ","End":"01:15.260","Text":"that equals to 0.6."},{"Start":"01:15.260 ","End":"01:20.660","Text":"That\u0027s the probability of a person not having blood type A."},{"Start":"01:20.660 ","End":"01:23.210","Text":"Now, how many Bernoulli trials do I have,"},{"Start":"01:23.210 ","End":"01:24.980","Text":"or how many people did I sample?"},{"Start":"01:24.980 ","End":"01:26.210","Text":"I sampled 4 people,"},{"Start":"01:26.210 ","End":"01:29.060","Text":"so n here, that equals to 4."},{"Start":"01:29.060 ","End":"01:36.985","Text":"Now, let\u0027s define X as the number of people who have blood type A."},{"Start":"01:36.985 ","End":"01:39.530","Text":"Now, this is what I\u0027m interested in."},{"Start":"01:39.530 ","End":"01:43.850","Text":"X then I know has a binomial distribution,"},{"Start":"01:43.850 ","End":"01:48.005","Text":"where N equals to 4 and the probability of success,"},{"Start":"01:48.005 ","End":"01:51.180","Text":"that equals to 0.4."},{"Start":"01:51.180 ","End":"01:56.855","Text":"Now we\u0027re looking for the probability that exactly one of them,"},{"Start":"01:56.855 ","End":"02:03.190","Text":"1 of these 4 people has blood type A."},{"Start":"02:03.190 ","End":"02:08.760","Text":"We\u0027re looking for the probability of X being equal to 1."},{"Start":"02:10.150 ","End":"02:15.890","Text":"Let\u0027s just recall the probability of X being equal to k,"},{"Start":"02:15.890 ","End":"02:18.455","Text":"well, that equals to n over k,"},{"Start":"02:18.455 ","End":"02:20.555","Text":"p to the power of k,"},{"Start":"02:20.555 ","End":"02:25.325","Text":"q to the power of n minus k. In our case,"},{"Start":"02:25.325 ","End":"02:27.890","Text":"the probability of X equaling 1,"},{"Start":"02:27.890 ","End":"02:29.660","Text":"well that equals to n over k,"},{"Start":"02:29.660 ","End":"02:31.505","Text":"that\u0027s 4 over 1,"},{"Start":"02:31.505 ","End":"02:34.000","Text":"p, 0.4,"},{"Start":"02:34.000 ","End":"02:37.760","Text":"to the power of k to the power of 1 times q,"},{"Start":"02:37.760 ","End":"02:41.810","Text":"0.6, to the power of n minus k and minus k,"},{"Start":"02:41.810 ","End":"02:43.850","Text":"that\u0027s 4 minus 1, that\u0027s 3."},{"Start":"02:43.850 ","End":"02:53.825","Text":"Now that equals to 0.432 and that does not equal to 0.4."},{"Start":"02:53.825 ","End":"02:57.960","Text":"Therefore, this assertion is not true."}],"ID":13278}],"Thumbnail":null,"ID":245057}]

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