Partitions Law of Total Probability Bayes’ Rule

Partitions Example 1 (From on-line Text). Your retail business is considering holding a sidewalk sale promotion next Saturday. Past experience indicates that the probability of a successful sale is 60%, if it does not rain. This drops to 30% if it does rain on Saturday. A phone call to the weather bureau finds an estimated probability of 20% for rain. What is the probability that you have a successful sale?

Partitions Let R be the event that it rains next Saturday and let N be the event that it does not rain next Saturday. Let A be the event that your sale is successful and let U be the event that your sale is unsuccessful. We are given that P(A|N) = 0.6 and P(A|R) = 0.3. The weather forecast states that P(R) = 0.2. Our goal is to compute P(A).

Partitions Conditional Probability Probability Event Probability Bayes’, Partitions Partitions Conditional Probability Probability Event Probability Saturday R N A R  A 0.20.3 = 0.06 A N  A 0.80.6 = 0.48 U R  U 0.20.7 = 0.14 U N  U 0.80.4 = 0.32 0.2 0.8 0.7 0.3 0.6 0.4

Partitions S=RN R N A P(A) = P(R  A) + P(N  A) = 0.06 + 0.48 = 0.54

Partition S B1 B2 B3 A

Partition 1. Bi  Bj = , unless i = j. 2. B1  B2    Bn = S. Let the events B1, B2, , Bn be non-empty subsets of a sample space S for an experiment. The Bi’s are a partition of S if the intersection of any two of them is empty, and if their union is S. This may be stated symbolically in the following way. 1. Bi  Bj = , unless i = j. 2. B1  B2    Bn = S.

Law of Total Probability Let the events B1, B2, , Bn partition the finite discrete sample space S for an experiment and let A be an event defined on S.

Law of Total Probability

Bayes’ Theorem Suppose that the events B1, B2, B3, . . . , Bn partition the sample space S for some experiment and that A is an event defined on S. For any integer, k, such that we have

Example 1 Scenario: Jar I contains 4 red balls and jar II contains 3 red balls and 2 yellow balls. Experiment: Choose a jar at random and from this jar select a ball at random Events: R = selected ball is red Y = selected ball is yellow J1 = Jar I is selected J2 = Jar II is selected What is P(Y|J2)? What is P(J2|Y)? What is P(J2|R)?

Example 2 (From on-line Text) All tractors made by a company are produced on one of three assembly lines, named Red, White, and Blue. The chances that a tractor will not start when it rolls off of a line are 6%, 11%, and 8% for lines Red, White, and Blue, respectively. 48% of the company’s tractors are made on the Red line and 31% are made on the Blue line. What fraction of the company’s tractors do not start when they roll off of an assembly line?

Tractor Example R Tractor W B S 0.4512 S 0.1869 S 0.2852 N 0.0288 0.0288 N 0.0231 0.0231 N 0.0248 0.0248 1.0000 0.0767  0.08 0.94 0.48 0.06 0.89 0.21 0.11 0.31 0.92 0.08

Example 2 - continued Given that a tractor does not start, what is the probability that it came from the Blue assembly line?

Example 3 Records show that when drilling for oil, the probability of a successful strike is 0.1. However, it has been observed that if there is oil, then the probability is 0.6 that there is permeable, porous sedimentary rock present. Records show that when there is no oil, the probability is 0.3 that such rock formations are present. What is the probability of oil beneath permeable, porous sedimentary rock?

Example 4 Thirty percent of the population have a certain disease. Of those that have the disease, 89% will test positive for the disease. Of those that do not have the disease, 5% will test positive. What is the probability that a person has the disease, given that they test positive for the disease?

Example 5 Given the following events What is ? A: Client has 10 yrs experience in business B: Client has a graduate degree C: Current economic conditions are normal S: A proposed workout is successful F: A proposed workout agreement fails What is ?