STAT 430/510 Probability Lecture 5: Conditional Probability and Bayes Rule
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1 STAT 430/510 Probability Lecture 5: Conditional Probability and Bayes Rule Pengyuan (Penelope) Wang May 26, 2011
2 Review Sample Spaces and Events Axioms and Properties of Probability
3 Conditional Probability The Probability of an event measures how often it will occur. A conditional probability predicts how often an event will occur under specified conditions. Notation: P(E F) represents the conditional probability that event E occurs, given that event F has occurred.
4 A coin is flipped twice. Assuming that all four points in the sample space S = {(H, H), (H, T ), (T, H), (T, T )} are equally likely, what is the conditional probability that both flips land on heads, given that (a) the first flip lands on heads? (b) at least one flip lands on heads? Intuitively, what is the answer?
5 Definition Definition If P(F) > 0, then P(E F) = P(EF) P(F) The "condition" F contains partial knowledge.
6 A coin is flipped twice. Assuming that all four points in the sample space S = {(H, H), (H, T ), (T, H), (T, T )} are equally likely, what is the conditional probability that both flips land on heads, given that (a) the first flip lands on heads? (b) at least one flip lands on heads? Compute with the formula P(E F) = P(EF) P(F ). Steps of thinking: What is E? What is F? And then what is EF?
7 A coin is flipped twice. Assuming that all four points in the sample space S = {(H, H), (H, T ), (T, H), (T, T )} are equally likely, what is the conditional probability that both flips land on heads, given that (a) the first flip lands on heads? (b) at least one flip lands on heads? Compute with the formula P(E F) = P(EF) P(F ). Steps of thinking: What is E? What is F? And then what is EF? What is P(EF), P(F)?
8 A coin is flipped twice. Assuming that all four points in the sample space S = {(H, H), (H, T ), (T, H), (T, T )} are equally likely, what is the conditional probability that both flips land on heads, given that (a) the first flip lands on heads? (b) at least one flip lands on heads? Compute with the formula P(E F) = P(EF) P(F ). Steps of thinking: What is E? What is F? And then what is EF? What is P(EF), P(F)? Then usep(e F) = P(EF) P(F )?
9 Multiplication Rule Conditional probability: P(E F) = P(EF) P(F) So, P(EF) = P(E)P(F E) = P(F)P(E F)
10 Sarah is undecided as to whether to take a French course or a chemistry course. She estimates that her probability of receiving an A grade would be 0.5 in a French course and 2/3 in a chemistry course. If Sarah decides to base her decision on the flip of a fair coin, what is the probability that she gets an A in chemistry?
11 Sarah is undecided as to whether to take a French course or a chemistry course. She estimates that her probability of receiving an A grade would be 0.5 in a French course and 2/3 in a chemistry course. If Sarah decides to base her decision on the flip of a fair coin, what is the probability that she gets an A in chemistry? Steps of thinking: Setup A and B.
12 Sarah is undecided as to whether to take a French course or a chemistry course. She estimates that her probability of receiving an A grade would be 0.5 in a French course and 2/3 in a chemistry course. If Sarah decides to base her decision on the flip of a fair coin, what is the probability that she gets an A in chemistry? Steps of thinking: Setup A and B. Let event A to be she chooses chemistry, event B to be she gets an A in chemistry.
13 Sarah is undecided as to whether to take a French course or a chemistry course. She estimates that her probability of receiving an A grade would be 0.5 in a French course and 2/3 in a chemistry course. If Sarah decides to base her decision on the flip of a fair coin, what is the probability that she gets an A in chemistry? Steps of thinking: Setup A and B. Let event A to be she chooses chemistry, event B to be she gets an A in chemistry. What is P(A), P(B A)?
14 Sarah is undecided as to whether to take a French course or a chemistry course. She estimates that her probability of receiving an A grade would be 0.5 in a French course and 2/3 in a chemistry course. If Sarah decides to base her decision on the flip of a fair coin, what is the probability that she gets an A in chemistry? Steps of thinking: Setup A and B. Let event A to be she chooses chemistry, event B to be she gets an A in chemistry. What is P(A), P(B A)? So P(AB) = P(A)P(B A).
15 Multiplication Rule More general: Multiplication Rule: For any events E 1,, E n, P(E 1 E n ) = P(E 1 )P(E 2 E 1 )P(E 3 E 1, E 2 ) P(E n E 1 E n 1 )
16 An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each. compute the probability that each pile has exactly 1 ace. Let event E 1 to the first person get exactly 1 ace, event E 2 to the second person get exactly 1 ace, event E 3 to the third person get exactly 1 ace,event E 4 to the fourth person get exactly 1 ace.
17 An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each. compute the probability that each pile has exactly 1 ace. Let event E 1 to the first person get exactly 1 ace, event E 2 to the second person get exactly 1 ace, event E 3 to the third person get exactly 1 ace,event E 4 to the fourth person get exactly 1 ace. What is P(E 1 ), P(E 2 E 1 ), P(E 3 E 1, E 2 ), and P(E 4 E 1, E 2, E 3 )?
18 An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each. compute the probability that each pile has exactly 1 ace. Let event E 1 to the first person get exactly 1 ace, event E 2 to the second person get exactly 1 ace, event E 3 to the third person get exactly 1 ace,event E 4 to the fourth person get exactly 1 ace. What is P(E 1 ), P(E 2 E 1 ), P(E 3 E 1, E 2 ), and P(E 4 E 1, E 2, E 3 )? So P(E 1 E 2 E 3 E 4 ) = P(E 1 )P(E 2 E 1 )P(E 3 E 1, E 2 )P(E 4 E 1, E 2, E 3 ).
19 The Law of Total Probability Suppose that the events A 1,, A n are a partition of the sample space S, that is, A 1,, A n are mutually exclusive and A i = S. Then for any event B, P(B) = P(B A 1 )P(A 1 ) + + P(B A n )P(A n )
20 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. If we assume that 30 percent of the population is accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy?
21 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. If we assume that 30 percent of the population is accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy? Steps of thinking: What is B? What is A 1, etc.?
22 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. If we assume that 30 percent of the population is accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy? Steps of thinking: What is B? What is A 1, etc.? What is P(B A 1 ), etc.?
23 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. If we assume that 30 percent of the population is accident prone, what is the probability that a new policyholder will have an accident within a year of purchasing a policy? Steps of thinking: What is B? What is A 1, etc.? What is P(B A 1 ), etc.? Then what is P(B)?
24 Bayes s Formula If P(B) > 0, then P(A B) = P(AB) P(B) = P(B A)P(A) P(B A)P(A) + P(B A c )P(A c ) If P(B) > 0 and A 1,, A n are a partition of the sample space S, then P(A i B) = P(A ib) P(B) = P(B A i )P(A i ) n j=1 P(B A j)p(a j )
25 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone? Steps of thinking: What is B? What is A 1, etc.?
26 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone? Steps of thinking: What is B? What is A 1, etc.? What is P(B A 1 ), etc.?
27 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone? Steps of thinking: What is B? What is A 1, etc.? What is P(B A 1 ), etc.? Then use the Bayes s Formula.
28 An insurance company believes that people can be divided into two classes: those who are accident prone and those who are not. The company s statistics show that an accident-prone person will have an accident at some time within a fixed 1-year period with probability 0.4, whereas this probability decreases to 0.2 for a person who is not accident prone. Suppose that a new policyholder has an accident within a year of purchasing a policy. What is the probability that he or she is accident prone? Steps of thinking: What is B? What is A 1, etc.? What is P(B A 1 ), etc.? Then use the Bayes s Formula. Or you can just compute P(B) and then use the formula P(A B) = P(AB) P(B).
29 A lab test yields 2 possible results: positive or negative. 99% of a particular disease will produce a positive result, but 2% of people without the disease will also produce a positive result. Suppose that 0.1% of the population actually has the disease. What is the probability that a person chosen at random will have the disease, given that the person s blood yields a positive result?
30 D=disease, +=positive test result Want P(D +)=? P(D)=0.001, P(D c ) = P(+ D)=0.99, and P(+ D c )=0.02
31 D=disease, +=positive test result Want P(D +)=? P(D)=0.001, P(D c ) = P(+ D)=0.99, and P(+ D c )=0.02 Applying Bayes Rule, P(D +) = P(D, +) P(+) = = P(+ D)P(D) P(+ D)P(D) + P(+ D c )P(D c ) = 4.7%
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