WEBVTT 00:00:02.530 --> 00:00:07.220 You've tested positive for a rare and deadly cancer that afflicts 1 out of 1000 people, 00:00:07.220 --> 00:00:11.980 based on a test that is 99% accurate. What are the chances that you actually have the 00:00:11.980 --> 00:00:17.560 cancer? By the end of this video, you'll be able to answer this question! 00:00:17.560 --> 00:00:22.510 This video is part of the Probability and Statistics video series. Many natural and 00:00:22.510 --> 00:00:28.260 social phenomena are probabilistic in nature. Engineers, scientists, and policymakers often 00:00:28.260 --> 00:00:31.930 use probability to model and predict system behavior. 00:00:31.930 --> 00:00:38.469 Hi, my name is Sam Watson, and I'm a graduate student in mathematics at MIT. 00:00:38.469 --> 00:00:43.679 Before watching this video, you should be familiar with basic probability vocabulary 00:00:43.679 --> 00:00:46.989 and the definition of conditional probability. 00:00:46.989 --> 00:00:51.219 After watching this video, you'll be able to: Calculate the conditional probability 00:00:51.219 --> 00:00:56.269 of a given event using tables and trees; and Understand how conditional probability can 00:00:56.269 --> 00:01:03.269 be used to interpret medical diagnoses. 00:01:04.479 --> 00:01:11.479 Suppose that in front of you are two bowls, labeled A and B. Each bowl contains five marbles. 00:01:11.600 --> 00:01:18.240 Bowl A has 1 blue and 4 yellow marbles. Bowl B has 3 blue and 2 yellow marbles. 00:01:18.240 --> 00:01:23.229 Now choose a bowl at random and draw a marble uniformly at random from it. Based on your 00:01:23.229 --> 00:01:28.200 existing knowledge of probability, how likely is it that you pick a blue marble? How about 00:01:28.200 --> 00:01:33.109 a yellow marble? 00:01:33.109 --> 00:01:40.109 Out of the 10 marbles you could choose from, 4 are blue. So the probability of choosing a blue 00:01:55.130 --> 00:01:58.020 marble is 4 out of 10. 00:01:58.020 --> 00:02:03.109 There are 6 yellow marbles out of 10 total, so the probability of choosing yellow is 6 00:02:03.109 --> 00:02:03.270 out of 10. 00:02:03.270 --> 00:02:04.109 When the number of possible outcomes is finite, and all events are equally likely, the probability 00:02:04.109 --> 00:02:05.009 of one event happening is the number of favorable outcomes divided by the total number of possible 00:02:05.009 --> 00:02:05.070 outcomes. 00:02:05.070 --> 00:02:09.470 What if you must draw from Bowl A? What's the probability of drawing a blue marble, 00:02:09.470 --> 00:02:16.470 given that you draw from Bowl A? 00:02:18.239 --> 00:02:25.239 Let's go back to the table and consider only Bowl A. Bowl A contains 5 marbles of which 00:02:25.599 --> 00:02:31.300 1 is blue, so the probability of picking a blue one is 1 in 5. 00:02:31.300 --> 00:02:36.610 Notice the probability has changed. In the first scenario, the sample space consists 00:02:36.610 --> 00:02:42.329 of all 10 marbles, because we are free to draw from both bowls. 00:02:42.329 --> 00:02:48.220 In the second scenario, we are restricted to Bowl A. Our new sample space consists of 00:02:48.220 --> 00:02:55.220 only the five marbles in Bowl A. We ignore these marbles in Bowl. 00:02:55.670 --> 00:03:00.909 Restricting our attention to a specific set of outcomes changes the sample space, and 00:03:00.909 --> 00:03:07.730 can also change the probability of an event. This new probability is what we call a conditional 00:03:07.730 --> 00:03:08.370 In the previous example, we calculated the conditional probability of drawing a blue 00:03:08.370 --> 00:03:09.659 marble, given that we draw from Bowl A. 00:03:09.659 --> 00:03:14.510 This is standard notation for conditional probability. The vertical bar ( | ) is read 00:03:14.510 --> 00:03:21.510 as "given." The probability we are looking for precedes the bar, and the condition follows 00:03:25.099 --> 00:03:32.099 the bar. 00:03:32.299 --> 00:03:37.439 Now let's flip things around. Suppose someone picks a marble at random from either bowl 00:03:37.439 --> 00:03:43.939 A or bowl B and reveals to you that the marble drawn was blue. What is the probability that 00:03:43.939 --> 00:03:46.989 the blue marble came from Bowl A? 00:03:46.989 --> 00:03:52.159 In other words, what's the conditional probability that the marble was drawn from Bowl A, given 00:03:52.159 --> 00:03:59.159 that it is blue? Pause the video and try to work this out. 00:04:01.159 --> 00:04:06.400 Going back to the table, because we are dealing with the condition that the marble is blue, 00:04:06.400 --> 00:04:11.170 the sample space is restricted to the four blue marbles. 00:04:11.170 --> 00:04:17.290 Of these four blue marbles, one is in Bowl A, and each is equally likely to be drawn. 00:04:17.290 --> 00:04:22.320 Thus, the conditional probability is 1 out of 4. 00:04:22.320 --> 00:04:27.160 Notice that the probability of picking a blue marble given that the marble came from Bowl 00:04:27.160 --> 00:04:33.030 A is NOT equal to the probability that the marble came from Bowl A given that the marble 00:04:33.030 --> 00:04:40.030 was blue. Each has a different condition, so be careful not to mix them up! 00:04:42.919 --> 00:04:47.650 We've seen how tables can help us organize our data and visualize changes in the sample 00:04:47.650 --> 00:04:49.060 space. 00:04:49.060 --> 00:04:53.180 Let's look at another tool that is useful for understanding conditional probabilities 00:04:53.180 --> 00:04:55.870 - a tree diagram. 00:04:55.870 --> 00:05:02.870 Suppose we have a jar containing 5 marbles; 2 are blue and 3 are yellow. If we draw any 00:05:03.220 --> 00:05:08.280 one marble at random, the probability of drawing a blue marble is 2/5. 00:05:08.280 --> 00:05:14.280 Now, without replacing the first marble, draw a second marble from the jar. Given that the 00:05:14.280 --> 00:05:20.330 first marble is blue, is the probability of drawing a second blue marble still 2/5? 00:05:20.330 --> 00:05:27.250 NO, it isn't. Our sample space has changed. If a blue marble is drawn first, you are left 00:05:27.250 --> 00:05:31.569 with 4 marbles; 1 blue and 3 yellow. 00:05:31.569 --> 00:05:36.130 In other words, if a blue marble is selected first, the probability that you draw blue 00:05:36.130 --> 00:05:42.580 second is 1/4. And the probability you draw yellow second is 3/4. 00:05:42.580 --> 00:05:49.580 Now pause the video and determine the probabilities if the yellow marble is selected first instead. 00:05:54.659 --> 00:06:00.539 If a yellow marble is selected first, you are left with 2 yellow and 2 blue marbles. 00:06:00.539 --> 00:06:06.389 There is now a 2/4 chance of drawing a blue marble and a 2/4 chance of drawing a yellow 00:06:06.389 --> 00:06:08.060 marble. 00:06:08.060 --> 00:06:15.060 What we have drawn here is called a tree diagram. The probability assigned to the second branch 00:06:18.550 --> 00:06:21.660 denotes the conditional probability given that the first happened. 00:06:21.660 --> 00:06:23.139 Tree diagrams help us to visualize our sample space and reason out probabilities. 00:06:23.139 --> 00:06:27.500 We can answer questions like "What is the probability of drawing 2 blue marbles in a 00:06:27.500 --> 00:06:32.550 row?" In other words, what is the probability of drawing a blue marble first AND a blue 00:06:32.550 --> 00:06:34.479 marble second? 00:06:34.479 --> 00:06:39.880 This event is represented by these two branches in the tree diagram. 00:06:39.880 --> 00:06:46.880 We have a 2/5 chance followed by a 1/4 chance. We multiply these to get 2/20, or 1/10. The 00:06:48.849 --> 00:06:53.349 probability of drawing two blue marbles in a row is 1/10. 00:06:53.349 --> 00:06:58.729 Now you do it. Use the tree diagram to calculate the probabilities of the other possibilities: 00:06:58.729 --> 00:07:05.729 blue, yellow; yellow, blue; and yellow, yellow. 00:07:10.050 --> 00:07:16.750 The probabilities each work out to 3/10. The four probabilities add up to a total of 1, 00:07:16.750 --> 00:07:18.800 as they should. 00:07:18.800 --> 00:07:22.599 What if we don't care about the first marble? We just want to determine the probability 00:07:22.599 --> 00:07:26.330 that the second marble is yellow. 00:07:26.330 --> 00:07:30.699 Because it does not matter whether the first marble is blue or yellow, we consider both 00:07:30.699 --> 00:07:37.699 the blue, yellow, and the yellow, yellow paths. Adding the probabilities gives us 3/10 + 3/10, 00:07:38.099 --> 00:07:41.139 which works out to 3/5. 00:07:41.139 --> 00:07:45.190 Here's another interesting question. What is the probability that the first marble drawn 00:07:45.190 --> 00:07:48.819 is blue, given that the second marble drawn is yellow? 00:07:48.819 --> 00:07:54.050 Intuitively, this seems tricky. Pause the video and reason through the probability tree 00:07:54.050 --> 00:08:01.050 with a friend. 00:08:01.370 --> 00:08:05.680 Because we are conditioning on the event that the second marble drawn is yellow, our sample 00:08:05.680 --> 00:08:09.289 space is restricted to these two paths: P(blue, yellow) and P(yellow, yellow). 00:08:09.289 --> 00:08:14.690 Of these two paths, only the top one meets our criteria - that the blue marble is drawn 00:08:14.690 --> 00:08:16.759 first. 00:08:16.759 --> 00:08:21.919 We represent the probability as a fraction of favorable to possible outcomes. Hence, 00:08:21.919 --> 00:08:26.199 the probability that the first marble drawn is blue, given that the second marble drawn 00:08:26.199 --> 00:08:33.198 is yellow is 3/10 divided by (3/10 +3/10), which works out to 1/2. 00:08:33.450 --> 00:08:39.000 I hope you appreciate that tree diagrams and tables make these types of probability problems 00:08:39.000 --> 00:08:46.000 doable without having to memorize any formulas! 00:08:47.210 --> 00:08:51.830 Let's return to our opening question. Recall that you've tested positive for a cancer that 00:08:51.830 --> 00:08:56.780 afflicts 1 out of 1000 people, based on a test that is 99% accurate. 00:08:56.780 --> 00:09:02.950 More precisely, out of 100 test results, we expect about 99 correct results and only 1 00:09:02.950 --> 00:09:05.100 incorrect result. 00:09:05.100 --> 00:09:10.290 Since the test is highly accurate, you might conclude that the test is unlikely to be wrong, 00:09:10.290 --> 00:09:13.110 and that you most likely have cancer. 00:09:13.110 --> 00:09:19.320 But wait! Let's first use conditional probability to make sense of our seemingly gloomy diagnosis. 00:09:19.320 --> 00:09:20.850 Now pause the video and determine the probability that you have the cancer, given that you test 00:09:20.850 --> 00:09:20.940 positive. 00:09:20.940 --> 00:09:24.470 Let's use a tree diagram to help with our calculations. 00:09:24.470 --> 00:09:30.650 The first branch of the tree represents the likelihood of cancer in the general population. 00:09:30.650 --> 00:09:36.910 The probability of having the rare cancer is 1 in 1000, or 0.001. The probability of 00:09:36.910 --> 00:09:41.790 having no cancer is 0.999. 00:09:41.790 --> 00:09:46.260 Let's extend the tree diagram to illustrate the possible results of the medical test that 00:09:46.260 --> 00:09:49.020 is 99% accurate. 00:09:49.020 --> 00:09:56.020 In the cancer population, 99% will test positive (correctly), but 1% will test negative (incorrectly). 00:09:57.150 --> 00:10:01.200 These incorrect results are called false negatives. 00:10:01.200 --> 00:10:07.520 In the cancer-free population, 99% will test negative (correctly), but 1% will test positive 00:10:07.520 --> 00:10:13.590 (incorrectly). These incorrect results are called false positives. 00:10:13.590 --> 00:10:19.270 Given that you test positive, our sample space is now restricted to only the population that 00:10:19.270 --> 00:10:24.920 test positive. This is represented by these two paths. 00:10:24.920 --> 00:10:30.760 The top path shows the probability you have the cancer AND test positive. The lower path 00:10:30.760 --> 00:10:37.440 shows the probability that you don't have cancer AND still test positive. 00:10:37.440 --> 00:10:44.440 The probability that you actually do have the cancer, given that you test positive, is (0.001*0.99)/((0.001*0.99)+(0.999*0.01)), 00:10:55.720 --> 00:11:01.150 which works out to about 0.09 - less than 10%! 00:11:01.150 --> 00:11:06.880 The error rate of the test is only 1 percent, but the chance of a misdiagnosis is more than 00:11:06.880 --> 00:11:13.550 90%! Chances are pretty good that you do not actually have cancer, despite the rather accurate 00:11:13.550 --> 00:11:16.760 test. Why is this so? 00:11:16.760 --> 00:11:22.440 The accuracy of the test actually reflects the conditional probability that one tests 00:11:22.440 --> 00:11:25.070 positive, given that one has cancer. 00:11:25.070 --> 00:11:30.520 But in practice, what you want to know is the conditional probability that you have 00:11:30.520 --> 00:11:37.520 cancer, given that you test positive! These probabilities are NOT the same! 00:11:37.520 --> 00:11:42.550 Whenever we take medical tests, or perform experiments, it is important to understand 00:11:42.550 --> 00:11:47.260 what events our results are conditioned on, and how that might affect the accuracy of 00:11:47.260 --> 00:11:53.180 our conclusions. 00:11:53.180 --> 00:11:57.180 In this video, you've seen that conditional probability must be used to understand and 00:11:57.180 --> 00:12:02.810 predict the outcomes of many events. You've also learned to evaluate and manage conditional 00:12:02.810 --> 00:12:06.830 probabilities using tables and trees. 00:12:06.830 --> 00:12:11.310 We hope that you will now think more carefully about the probabilities you encounter, and 00:12:11.310 --> 00:12:14.200 consider how conditioning affects their interpretation.