1.6/1.7 - Conditional Probability and Bayes Theorem

Transcription

1 1.6/1.7 - Conditional Probability and Bayes Theorem Math Blake Boudreaux Department of Mathematics Texas A&M University February 1, 2018 Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

2 Introduction Roll two dice. If their sum is a two-digit number you win. What is the probability of winning? Solution: Suppose you rolled one of the dice first and rolled a six. What is the probability of winning now? Solution: Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

3 Definition Definition The conditional probability of event E given event F has occurred is defined by P(E F ) = P(E F ). P(F ) For example: The idea is to reduce the sample space. Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

4 Theorem Theorem If E and F are two events in a sample space S with P(E) > 0 and P(F ) > 0, then P(E F ) = P(F )P(E F ) = P(E)P(F E). Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

5 Example You and your friend eat M&Ms by color. As a result, your bag is down to 4 red and 12 green M&Ms. You select an M&M. It looks bad, so you set it aside and select another one. a. What is the probability this M&M (the second one selected) is a red one? b. What is the probability that both M&M s selected are red? c. Using the definition of conditional probability, find the probability the first M&M selected is red given the second M&M selected is red. Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

6 Theorem Part (c) of the previous is an example of Bayes Theorem given the results of what happened second, find the probability of what happened first to cause it. Theorem (Bayes Theorem) Let E 1, E 2, E 3,..., E n be mutually exclusive events in a sample space S with E 1 E 2 E 3 E n = S. For any event F, P(E i F ) = P(E i ) P(F E i ) P(E 1 ) P(F E 1 ) + + P(E n ) P(F E n ). You are not required to memorize this formula. Use tree diagrams. Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

7 Example Suppose instead you put the first M&M back in the bag (because it wasn t the color you wanted) and select another one. Now what is the probability the second M&M selected is a red one? Solution: Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

8 Definition Notice that in the first example, the probabilities in the tree diagram changed after each selection, while in the second example, they did not. The second example gives us examples of independent events. Definition Events E and F are said to be independent if P(E F ) = P(E) and P(F E) = P(F ). Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

9 Theorem Using the product rule for probability, we see the following: Theorem Let E and F be two events with P(E) > 0 and P(F ) > 0. Then E and F are independent if, and only if, P(E F ) = P(E)P(F ). Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

10 Example In the experiment that you roll two dice, let E be the event that the sum is a two-digit number, and let F be the event that a four is rolled on the first die. Determine if E and F are independent. Solution: Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

11 Example Each new battery independently has a 99% chance of working properly (not being defective). Right before your first exam, you replace the 4 batteries in your calculator. a. What is the probability that at least one of them is bad? b. What is the probability that exactly one of them is bad? Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

12 Example A medical test to determine gluten sensitivity will correctly detect the allergies 99% of the time, but also give a false positive result 5% of the time. It is believed that 6% of the population has a gluten sensitivity. If you take the test and get a positive result, what is the probability that you have gluten sensitivity? Solution: Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13

13 Example Suppose after testing positive you take a second test with the same accuracy statistics as the first. If the second test is positive, what is the probability that you have gluten sensitivity? Solution: Blake Boudreaux (Texas A&M University) 1.6/1.7 - Conditional Probability and Bayes Theorem February 1, / 13