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Conditional Probability

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12:28   |   Jan 21, 2014

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Conditional Probability
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  • You've tested positive for a rare and deadly cancer that afflicts 1 out of 1000 people,
  • based on a test that is 99% accurate. What are the chances that you actually have the
  • cancer? By the end of this video, you'll be able to answer this question!
  • This video is part of the Probability and Statistics video series. Many natural and
  • social phenomena are probabilistic in nature. Engineers, scientists, and policymakers often
  • use probability to model and predict system behavior.
  • Hi, my name is Sam Watson, and I'm a graduate student in mathematics at MIT.
  • Before watching this video, you should be familiar with basic probability vocabulary
  • and the definition of conditional probability.
  • After watching this video, you'll be able to: Calculate the conditional probability
  • of a given event using tables and trees; and Understand how conditional probability can
  • be used to interpret medical diagnoses.
  • Suppose that in front of you are two bowls, labeled A and B. Each bowl contains five marbles.
  • Bowl A has 1 blue and 4 yellow marbles. Bowl B has 3 blue and 2 yellow marbles.
  • Now choose a bowl at random and draw a marble uniformly at random from it. Based on your
  • existing knowledge of probability, how likely is it that you pick a blue marble? How about
  • a yellow marble?
  • Out of the 10 marbles you could choose from, 4 are blue. So the probability of choosing a blue
  • marble is 4 out of 10.
  • There are 6 yellow marbles out of 10 total, so the probability of choosing yellow is 6
  • out of 10.
  • When the number of possible outcomes is finite, and all events are equally likely, the probability
  • of one event happening is the number of favorable outcomes divided by the total number of possible
  • outcomes.
  • What if you must draw from Bowl A? What's the probability of drawing a blue marble,
  • given that you draw from Bowl A?
  • Let's go back to the table and consider only Bowl A. Bowl A contains 5 marbles of which
  • 1 is blue, so the probability of picking a blue one is 1 in 5.
  • Notice the probability has changed. In the first scenario, the sample space consists
  • of all 10 marbles, because we are free to draw from both bowls.
  • In the second scenario, we are restricted to Bowl A. Our new sample space consists of
  • only the five marbles in Bowl A. We ignore these marbles in Bowl.
  • Restricting our attention to a specific set of outcomes changes the sample space, and
  • can also change the probability of an event. This new probability is what we call a conditional
  • In the previous example, we calculated the conditional probability of drawing a blue
  • marble, given that we draw from Bowl A.
  • This is standard notation for conditional probability. The vertical bar ( | ) is read
  • as "given." The probability we are looking for precedes the bar, and the condition follows
  • the bar.
  • Now let's flip things around. Suppose someone picks a marble at random from either bowl
  • A or bowl B and reveals to you that the marble drawn was blue. What is the probability that
  • the blue marble came from Bowl A?
  • In other words, what's the conditional probability that the marble was drawn from Bowl A, given
  • that it is blue? Pause the video and try to work this out.
  • Going back to the table, because we are dealing with the condition that the marble is blue,
  • the sample space is restricted to the four blue marbles.
  • Of these four blue marbles, one is in Bowl A, and each is equally likely to be drawn.
  • Thus, the conditional probability is 1 out of 4.
  • Notice that the probability of picking a blue marble given that the marble came from Bowl
  • A is NOT equal to the probability that the marble came from Bowl A given that the marble
  • was blue. Each has a different condition, so be careful not to mix them up!
  • We've seen how tables can help us organize our data and visualize changes in the sample
  • space.
  • Let's look at another tool that is useful for understanding conditional probabilities
  • - a tree diagram.
  • Suppose we have a jar containing 5 marbles; 2 are blue and 3 are yellow. If we draw any
  • one marble at random, the probability of drawing a blue marble is 2/5.
  • Now, without replacing the first marble, draw a second marble from the jar. Given that the
  • first marble is blue, is the probability of drawing a second blue marble still 2/5?
  • NO, it isn't. Our sample space has changed. If a blue marble is drawn first, you are left
  • with 4 marbles; 1 blue and 3 yellow.
  • In other words, if a blue marble is selected first, the probability that you draw blue
  • second is 1/4. And the probability you draw yellow second is 3/4.
  • Now pause the video and determine the probabilities if the yellow marble is selected first instead.
  • If a yellow marble is selected first, you are left with 2 yellow and 2 blue marbles.
  • There is now a 2/4 chance of drawing a blue marble and a 2/4 chance of drawing a yellow
  • marble.
  • What we have drawn here is called a tree diagram. The probability assigned to the second branch
  • denotes the conditional probability given that the first happened.
  • Tree diagrams help us to visualize our sample space and reason out probabilities.
  • We can answer questions like "What is the probability of drawing 2 blue marbles in a
  • row?" In other words, what is the probability of drawing a blue marble first AND a blue
  • marble second?
  • This event is represented by these two branches in the tree diagram.
  • We have a 2/5 chance followed by a 1/4 chance. We multiply these to get 2/20, or 1/10. The
  • probability of drawing two blue marbles in a row is 1/10.
  • Now you do it. Use the tree diagram to calculate the probabilities of the other possibilities:
  • blue, yellow; yellow, blue; and yellow, yellow.
  • The probabilities each work out to 3/10. The four probabilities add up to a total of 1,
  • as they should.
  • What if we don't care about the first marble? We just want to determine the probability
  • that the second marble is yellow.
  • Because it does not matter whether the first marble is blue or yellow, we consider both
  • the blue, yellow, and the yellow, yellow paths. Adding the probabilities gives us 3/10 + 3/10,
  • which works out to 3/5.
  • Here's another interesting question. What is the probability that the first marble drawn
  • is blue, given that the second marble drawn is yellow?
  • Intuitively, this seems tricky. Pause the video and reason through the probability tree
  • with a friend.
  • Because we are conditioning on the event that the second marble drawn is yellow, our sample
  • space is restricted to these two paths: P(blue, yellow) and P(yellow, yellow).
  • Of these two paths, only the top one meets our criteria - that the blue marble is drawn
  • first.
  • We represent the probability as a fraction of favorable to possible outcomes. Hence,
  • the probability that the first marble drawn is blue, given that the second marble drawn
  • is yellow is 3/10 divided by (3/10 +3/10), which works out to 1/2.
  • I hope you appreciate that tree diagrams and tables make these types of probability problems
  • doable without having to memorize any formulas!
  • Let's return to our opening question. Recall that you've tested positive for a cancer that
  • afflicts 1 out of 1000 people, based on a test that is 99% accurate.
  • More precisely, out of 100 test results, we expect about 99 correct results and only 1
  • incorrect result.
  • Since the test is highly accurate, you might conclude that the test is unlikely to be wrong,
  • and that you most likely have cancer.
  • But wait! Let's first use conditional probability to make sense of our seemingly gloomy diagnosis.
  • Now pause the video and determine the probability that you have the cancer, given that you test
  • positive.
  • Let's use a tree diagram to help with our calculations.
  • The first branch of the tree represents the likelihood of cancer in the general population.
  • The probability of having the rare cancer is 1 in 1000, or 0.001. The probability of
  • having no cancer is 0.999.
  • Let's extend the tree diagram to illustrate the possible results of the medical test that
  • is 99% accurate.
  • In the cancer population, 99% will test positive (correctly), but 1% will test negative (incorrectly).
  • These incorrect results are called false negatives.
  • In the cancer-free population, 99% will test negative (correctly), but 1% will test positive
  • (incorrectly). These incorrect results are called false positives.
  • Given that you test positive, our sample space is now restricted to only the population that
  • test positive. This is represented by these two paths.
  • The top path shows the probability you have the cancer AND test positive. The lower path
  • shows the probability that you don't have cancer AND still test positive.
  • 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)),
  • which works out to about 0.09 - less than 10%!
  • The error rate of the test is only 1 percent, but the chance of a misdiagnosis is more than
  • 90%! Chances are pretty good that you do not actually have cancer, despite the rather accurate
  • test. Why is this so?
  • The accuracy of the test actually reflects the conditional probability that one tests
  • positive, given that one has cancer.
  • But in practice, what you want to know is the conditional probability that you have
  • cancer, given that you test positive! These probabilities are NOT the same!
  • Whenever we take medical tests, or perform experiments, it is important to understand
  • what events our results are conditioned on, and how that might affect the accuracy of
  • our conclusions.
  • In this video, you've seen that conditional probability must be used to understand and
  • predict the outcomes of many events. You've also learned to evaluate and manage conditional
  • probabilities using tables and trees.
  • We hope that you will now think more carefully about the probabilities you encounter, and
  • consider how conditioning affects their interpretation.

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Description

MIT RES.TLL-004 Concept Vignettes
View the complete course: http://ocw.mit.edu/RES-TLL-004F13
Instructor: Sam Watson

This video provides an introduction to conditional probability and its calculations, as well as how it can be used to interpret medical diagnoses.

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
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