PHP 2510 Probability Part 3. Calculate probability by counting (cont ) Conditional probability. Law of total probability. Bayes Theorem.

Transcription

1 PHP 2510 Probability Part 3 Calculate probability by counting (cont ) Conditional probability Law of total probability Bayes Theorem Independence PHP 2510 Sept 22,

2 Last time... Count the number of ways to randomly sample r subjects (e.g. r labeled balls) Context: From a finite population (from an urn containing n balls) without replacement. If ordering is relevant (permutations), n (n 1)... (n r + 1) = n! (n r)!. If ordering does not matter (combinations), ( ) n! n (n r)!r! =. r PHP 2510 Sept 22,

3 Capture-recapture methods: Example 1 Suppose there are 100 bobcats in a certain state park. A scientist captures 10 of them, tags them, and releases them back to their habitat. A week later, he recaptures 20. What is the probability that 4 of the recaptured bobcats 20 have tags? Denominator: how many ways to draw the 20 bobcats from the population of 100? Numerator: In the recaptured sample of 20, how many ways to 1. Draw 4 tagged bobcats from the 10 that have tags 2. Draw 16 non-tagged bobcats from the 90 that do not have tags Ans: ( 10 4 )( ( ) ).08 PHP 2510 Sept 22,

4 Capture-recapture methods: Example 2 The most common use of capture-recapture experiments is to estimate a population size. In the previous example, assume the population size is unknown; denote it by N. The biologist captures 10 bobcats and tags them. A week later he captures 20 and finds that 4 are tagged. To find the population size, we can find the value of N that, in some sense, is the most consistent with the observed data. Hence we find N that maximizes the expression ( 10 )( N 10 ) ( N 20) In this case it turns out that N = 50 maximizes the probability. PHP 2510 Sept 22,

5 Counting possible allocations into several categories Property. The number of ways of grouping n objects into r classes is ( ) n! n 1! n 2! n r! = n n 1, n 2,..., n r This number is called the multinomial coefficient PHP 2510 Sept 22,

6 Example: Allocating committee members A committee of 7 members is divided into three subcommittees of size 4, 2 and 1. How many distinct ways can the subcommittees be formed? Numerator: number of ways of arranging the 7 members Denominator: number of ways to allocate the subcommittees Ans: ( ) 7 4, 2, 1 = 7! 4!2!1! = 105 PHP 2510 Sept 22,

7 Example: Genomic sequences A DNA molecule is a sequence of nucleotides labeled A, C, G, and T. The entire molecule can be millions of units long. Sometimes researchers are interested in the arrangement of nucleotides at a specific location on the DNA molecule. How many ways can the nucleotides {A, A, G, G, G, T, T, T, T } be arranged in a sequence that is 9 units long? Ans: ( ) 9 2, 3, 4 = 9! 2!3!4! = 1260 Why is this a problem of combinations, not permutations? PHP 2510 Sept 22,

8 Conditional probability Consider screening randomly-selected individuals for HIV. For each person we can define two events: the test status and the HIV status. H +, H = HIV status T +, T = test status PHP 2510 Sept 22,

9 In practice, we frequently will be interested in specific conditional probabilities: P (H + T + ) = prob of being HIV+, conditional on having a positive test P (T + H + ) = prob of having a positive test, conditional on being HIV+ PHP 2510 Sept 22,

10 Definition of conditional probability For any two events A and B, where P (B) > 0, the conditional probability of event A, given that event B already has occurred, is given by P (A B) = P (A B). P (B) The idea is that if we are given that event B occurred, the relevant sample space becomes B rather than the original sample size Ω. PHP 2510 Sept 22,

11 Example 1: HIV testing Recall the HIV testing situation. The sample space is Ω = {(H +, T + ), (H +, T ), (H, T + ), (H, T )} Suppose the probabilities are as follows: H + H T T Find the following: P (H + ), P (T + ), P (H + T + ), P (H + T + ) PHP 2510 Sept 22,

12 Computing conditional probabilities What is the probability of being HIV+, given that the test result is positive? (This is called the predictive value of a positive test). P (H + T + ) = =.71 What is the probability of being HIV, given that the test result is positive? Use the complement law. Ans:.29 What is the probability of testing positive, given that true HIV status is positive? (This is called sensitivity of a diagnostic test). P (T + H + ) = =.91 PHP 2510 Sept 22,

13 Multiplication law A direct result of this definition is the multiplication law, which can be used to find a joint probability of A B: P (A B) = P (B) P (A B) The multiplication law can be used to calculate a joint probability P (A B) when P (A B) and P (B) known. Example. An urn contains 5 red balls and 6 white balls. You select two balls at random, without replacement. What is the probability of selecting 2 red balls? Ans: =.18 PHP 2510 Sept 22,

14 Example. Suppose that if it is cloudy (B), the probability that it is raining (A) is 0.3, and that the probability that it is cloudy is P (B) is 0.2. What is the probability that it is cloudy and raining? Ans: Example. In a particular community, the prevalence of HIV is.10. Among those with HIV, it is found that 40% also have hepatitis C (HCV). What is the probability that a randomly selected person is infected with both HIV and HCV? Ans: PHP 2510 Sept 22,

15 Law of total probability Suppose the events B 1, B 2,..., B n are disjoint and exhaustive. By disjoint, we mean that B i B j = for any i j. By exhaustive, we mean that Then for any event A, P (B 1 B 2 B n ) = 1 P (A) = P (A B 1 ) + P (A B 2 ) + + P (A B n ) = n P (A B i ) i=1 Because P (A B i ) = P (A B i )P (B i ), we also have P (A) = P (A B 1 )P (B 1 ) + P (A B 2 )P (B 2 ) + + P (A B n )P (B n ) = n P (A B i )P (B i ) i=1 PHP 2510 Sept 22,

16 Law of total probability, in words Says that the probability of an event A is a weighted average of conditional probabilities taken over a set of mutually exclusive and exhaustive events. The conditional probabilities are P (A B i ) and they are weighted by P (B i ). PHP 2510 Sept 22,

17 Application of the law of total probability Consider MRI for breast CA, where M = 1, 2, 3 is true degree of malignancy (0=none, 1=benign, 2=malignant). Suppose prevalence of each underlying status, for the population being screened, is 0.80 (none), 0.15 (benign), 0.05 (malignant). Upon radiology scan, a biopsy is either ordered (B) or not ordered (B c ), depending on the judgment of the radiologist. Suppose a particular radiologist orders biopsies with the following probabilities: P (B M = 0) =.10 P (B M = 1) =.40 P (B M = 2) =.90 What is the probability that a patient undergoing breast MRI will be referred for a biopsy? Ans =.19 PHP 2510 Sept 22,

18 Bayes Theorem Let B 1,..., B n be a set of disjoint and exhaustive events. Then for some event A, the conditional probability P (B j A) is P (B j A) = P (B j A) P (A) = P (A B j )P (B j ) n i=1 P (A B i)p (B i ) Example. In the previous example, among those referred for biopsy, what is the probability of having a benign tumor? Ans:.42. What is the probability of having a malignant tumor? Ans:.24. PHP 2510 Sept 22,

19 Independent events Two events A and B are said to be independent if P (A B) = P (A)P (B) The intuition behind this can be seen as follows. We can say that A and B are independent if conditioning on B does not affect the probability of A occurring. PHP 2510 Sept 22,

20 In other word, P (A B) = P (A) and P (B A) = P (B). But we can rewrite P (A B) as P (A B) P (B), which implies the definition above. Note that A B = dose NOT imply that A and B are independent. PHP 2510 Sept 22,

21 Calculating probabilities for independent events Example 1. Suppose the population prevalence of TB (pulmonary tuberculosis) is.01. Two randomly-selected individuals are tested for TB. What is the probability that: (a) both have TB? (b) neither has TB? (c) exactly one has TB? Ans: (a).001; (b).9801; (c).0198 Example 2. In randomly selected cohabiting couples, it is found that the proportion of couples where both members have TB is.01. Is the presence of TB in one member of a couple independent of presence of TB in the other member? PHP 2510 Sept 22,

22 Example 3. A gene expression array has 10,000 cells that each measure RNA expression of individual genes. The false positive rate for any given cell i.e., the probability of indicating gene activity when it is in fact absent is 1 in The expression array is applied to a neutral medium, where it is known that none of the genes on the array will be active. What is the probability that the array will correctly indicate no activity for each of the 10,000 genes? Ans:.135 Example 4. Suppose that the probability of contracting HIV in one act of sexual intercourse is 1 in 500. If a person has 100 sexual encounters with an HIV-infected individual, what is the probability of contracting HIV? Ans:.181 PHP 2510 Sept 22,