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1 Math 141 Lecture 2: More Probability! Albyn Jones 1 1 Library jones/courses/141

2 Outline Law of total probability Bayes Theorem

3 the Multiplication Rule, again Recall that for any two events A and B: P(A B) = P(A B) P(B) Suppose there are several events B 1, B 2,... B k, the multiplication rule applies to each one: etc. P(A B 1 ) = P(A B 1 ) P(B 1 ) P(A B 2 ) = P(A B 2 ) P(B 2 )

4 A Partition Definition: A partition of the sample space Ω is a collection of disjoint events B 1, B 2,... B k whose union is Ω. Such a partition divides any set A into disjoint pieces: Ω A B 1 B 2 B 3 B 4 B 5 B 6

5 Application: The law of Total Probability Suppose we have a partition B 1, B 2,... B k, then any other event A is a union of its pieces: A = (A B 1 ) (A B 2 )... (A B k ) Those pieces are disjoint, so P(A) = P(A B 1 ) + P(A B 2 ) P(A B k ) Applying the multiplcation rule gives: P(A) = P(A B 1 )P(B 1 ) + P(A B 2 )P(B 2 ) P(A B k )P(B k )

6 Example Suppose we draw two cards from a well shuffled deck. One partition of the sample space is: A c 1 A 1 = {The first card is an Ace} = {The first card is not an Ace} It is often useful to look for partitions of the sample space that simplify a computation!

7 Example: The law of Total Probability Consider a well shuffled card deck. What is the probability the second card in the deck is an ace? Condition on whether the first card is an ace or not: P(A 2 ) = P(A 2 A 1 )P(A 1 ) + P(A 2 A c 1 )P(Ac 1 ) We can compute all of these terms easily: = P(A 2 ) = = (3 + 48) = = 4 52

8 Another Example: Mixtures Suppose we have two hats: one has 4 red balls and 6 green balls, the other has 6 red and 4 green. We toss a fair coin, if heads, pick a random ball from the first hat, if tails from the second. What is the probability of getting a red ball? P(Red) = P(Red H) P(H) + P(Red T ) P(T ) = = 1 2 The total probability of drawing a red ball is a weighted average of the two conditional probabilities, where the weights are the probabilities of each condition occuring.

9 Go Timbers! A soccer team wins 60% of its games when it scores the first goal, and 10% of its games when the opposing team scores first. If the team scores the first goal about 30% of the time, what fraction of the games does it win? Let W be the event that the team wins, and SF be the event that it scores first. Then the LTP says: P(W ) = P(W SF) P(SF ) + P(W SF c ) P(SF c ) or P(W ) = = = = 1 4

10 Bayes Theorem Bayes Theorem is really just the definition of conditional probability dressed up with the Law of Total Probability. Suppose we have a partition B 1, B 2,... B k, for which we know the probabilities P(A B i ), and we wish to compute P(B j A). P(B j A) = P(B j A) P(A) = P(A B j) P(B j ) P(A) Now use the LTP to compute the denominator: P(B j A) = P(A B j )P(B j ) P(A B 1 )P(B 1 ) + P(A B 2 )P(B 2 ) P(A B k )P(B k )

11 Example: Screening for a rare disease Suppose we have a blood test for a disease which has the following characteristics, where P = tests positive, and D = has the disease :

12 Example: Screening for a rare disease Suppose we have a blood test for a disease which has the following characteristics, where P = tests positive, and D = has the disease : Sensitivity: P(P D) =.99

13 Example: Screening for a rare disease Suppose we have a blood test for a disease which has the following characteristics, where P = tests positive, and D = has the disease : Sensitivity: P(P D) =.99 Specificity: P(P c D c ) =.99 Thus P(P D c ) =.01

14 Example: Screening for a rare disease Suppose we have a blood test for a disease which has the following characteristics, where P = tests positive, and D = has the disease : Sensitivity: P(P D) =.99 Specificity: P(P c D c ) =.99 Thus P(P D c ) =.01 Prevalence: P(D) =.01

15 Example: Screening for a rare disease Suppose we have a blood test for a disease which has the following characteristics, where P = tests positive, and D = has the disease : Sensitivity: P(P D) =.99 Specificity: P(P c D c ) =.99 Thus P(P D c ) =.01 Prevalence: P(D) =.01 Question: What is the probability that someone who tests positive actually has the disease?

16 Use Bayes! Clue: we want to get from P(P D) to P(D P). P(D P) = Substituting, we have P(D P) = P(P D)P(D) P(P D)P(D) + P(P D c )P(D c ) = 1 2 Roughly half of those who test positive do not have the disease!

17 What Happened? Imagine we have 1000 people: about 990 don t have the disease, and 10 do.

18 What Happened? Imagine we have 1000 people: about 990 don t have the disease, and 10 do. Of the 10 who have the disease, with high probability all 10 test positive.

19 What Happened? Imagine we have 1000 people: about 990 don t have the disease, and 10 do. Of the 10 who have the disease, with high probability all 10 test positive. Of the 990 who don t,1%, or about 10, test positive.

20 Example: Screening for a COMMON disease Using the same notation as before:

21 Example: Screening for a COMMON disease Using the same notation as before: Sensitivity: P(P D) =.99

22 Example: Screening for a COMMON disease Using the same notation as before: Sensitivity: P(P D) =.99 Specificity: P(P D c ) =.01

23 Example: Screening for a COMMON disease Using the same notation as before: Sensitivity: P(P D) =.99 Specificity: Prevalence: P(P D c ) =.01 P(D) =.10

24 Example: Screening for a COMMON disease Using the same notation as before: Sensitivity: P(P D) =.99 Specificity: Prevalence: P(P D c ) =.01 P(D) =.10 Question: What is the probability that someone who tests positive actually has the disease?

25 Use Bayes again! Same Clue: we want to get from P(P D) to P(D P). P(D P) = Substituting, we have P(D P) = P(P D)P(D) P(P D)P(D) + P(P D c )P(D c ) = The difference is due to the baseline rate P(D).

26 What Happened This Time? Imagine we have 1000 people: about 900 don t have the disease, and 100 do.

27 What Happened This Time? Imagine we have 1000 people: about 900 don t have the disease, and 100 do. Of the 100 who have the disease, almost all, or about 99 test positive.

28 What Happened This Time? Imagine we have 1000 people: about 900 don t have the disease, and 100 do. Of the 100 who have the disease, almost all, or about 99 test positive. Of the 900 who don t,1%, or about 9, test positive.

29 What Happened This Time? Imagine we have 1000 people: about 900 don t have the disease, and 100 do. Of the 100 who have the disease, almost all, or about 99 test positive. Of the 900 who don t,1%, or about 9, test positive. Hence the advice from Public Health officials: normally we don t bother screening the general population for rare diseases, but we might consider screening high risk groups.

30 A probability Puzzle: Let s Make a Deal! You re given the choice of three doors: Behind one door is a Valuable Prize; behind the others, Something Useless.

31 A probability Puzzle: Let s Make a Deal! You re given the choice of three doors: Behind one door is a Valuable Prize; behind the others, Something Useless. You pick a door, say No. 1.

32 A probability Puzzle: Let s Make a Deal! You re given the choice of three doors: Behind one door is a Valuable Prize; behind the others, Something Useless. You pick a door, say No. 1. The host, who knows what s behind the doors, opens another door, say No. 3, revealing Something Useless.

33 A probability Puzzle: Let s Make a Deal! You re given the choice of three doors: Behind one door is a Valuable Prize; behind the others, Something Useless. You pick a door, say No. 1. The host, who knows what s behind the doors, opens another door, say No. 3, revealing Something Useless. The host then asks you if you would like to switch to door No. 2.

34 A probability Puzzle: Let s Make a Deal! You re given the choice of three doors: Behind one door is a Valuable Prize; behind the others, Something Useless. You pick a door, say No. 1. The host, who knows what s behind the doors, opens another door, say No. 3, revealing Something Useless. The host then asks you if you would like to switch to door No. 2. What is the probability that the Valuable Prize is behind door No. 2?

35 Summary The Law of Total Probability For a partition B 1, B 2,... B k of Ω, the probability of any other event A is a weighted average of its conditional probabilities: P(A) = P(A B 1 )P(B 1 ) + P(A B 2 )P(B 2 ) P(A B k )P(B k ) Bayes Theorem For a partition B 1, B 2,... B k of Ω, we can reverse the order of conditioning on an event A to update our probabilities for events in the partition: P(B j A) = P(A B j )P(B j ) P(A B 1 )P(B 1 ) + P(A B 2 )P(B 2 ) P(A B k )P(B k )

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