Law of Total Probability and Bayes’ Rule

“Event-composition method” Understand the experiment and sample points. Using set notation, express the event of interest in terms of events for which the probability is known. Applying probability rules, combine the known probabilities to determine the probability of the specified event.

Problem 2.86 In a factory, 40% of items produced come from Line 1 and others from Line 2. Line 1 has a defect rate of 8%. Line 2 has a defect rate of 10%. For randomly selected item, find probability the item is not defective. A: the selected item is not defective

Problem 2.86 A: the selected item is not defective. B1: item came from Line 1. B2: item came from Line 2.

Problem 2.86 So we may write Since this is the union of disjoint sets, the Additive Law yields Or, in terms of conditional probabilities

The Decision Tree defective Line 1 not defective defective Line 2

Problem 2.94 Must find blood donor for an accident victim in the next 8 minutes or else… Checking blood types of potential donors requires 2 minutes each and may only be tested one at a time. 40% of the potential donors have the required blood type. What is the probability a satisfactory blood donor is identified in time to save the victim?

Finding a Donor A: blood donor is found within 8 minutes Some sample points: “B bad, G good” A = { (G), (B,G), (B,B,G), (B,B,B,G) } Let Ai: ith donor has correct blood type 4 mutually exclusive events

Finding a Donor Trials are independent and each P(Ai) = 0.40, and so

Finding a Donor saved! saved! saved! saved! too late!

Problem 2.96 Of 6 refrigerators, 2 don’t work. The refrigerators are tested one at a time. When tested, it’s clear whether it works! What is the probability the last defective refrigerator is found on the 4th test? What is the probability no more than 4 need to be tested to identify both defective refrigerators?

Problem 2.96 Given that exactly one defective refrigerator was found during the first 2 tests, what is the probability the other one is found on the 3rd or 4th test?

Partition of the Sample Space … B1 B2 Bk

Union of Disjoint Sets S A … B1 B2 Bk

Recall Problem 2.86 A: the selected item is not defective. B1: item came from Line 1. B2: item came from Line 2.

Law of Total Probability S A … B1 B2 Bk

Total Probability P(A|B1)P(B1) B1 P(A|B2)P(B2) B2 B3 P(A|B3)P(B3)

Bayes’ Theorem follows…

Bayes’ B1 B2 B3 P(A|B1)P(B1) P(A|B2)P(B2) P(A|B3)P(B3)

Making Resistors Three machines M1, M2, and M3 produce “1000-ohm” resistors. M1 produces 80% of resistors accurate to within 50 ohms, M2 produces 90% to within 50 ohms, and M3 produces 60% to within 50 ohms. Each hour, M1 produces 3000 resistors, M2 produces 4000, and M3 produces 3000. If all of the resistors are mixed together and shipped in a single container, what is the probability a selected resistor is accurate to within 50 ohms?

Making Resistors Define A: resistor is accurate to within 50 ohms. M1 produces 80% of resistors accurate to within 50 ohms, M2 produces 90% to within 50 ohms, and M3 produces 60% to within 50 ohms. Each hour, M1 produces 3000 resistors, M2 produces 4000, and M3 produces 3000.

Using Total Probability That is, 78 % are expected to be accurate to within 50 ohms.

The Tree (0.8)(0.3) M1 (0.9)(0.4) M2 M3 (0.6)(0.3)

…given it’s within 50 ohms… Determine the probability that, given a selected resistor is accurate to within 50 ohms, it was produced by M1. P( M1 | A) = ? Determine the probability that, given a selected resistor is accurate to within 50 ohms, it was produced by M3. P( M3 | A) = ?

Given A… (0.8)(0.3) M1 (0.9)(0.4) M2 M3 (0.6)(0.3)

Arthritis A test detects a particular type of arthritis for individuals over 50 years old. 10% of this age group suffers from this arthritis. For individuals in this age group known to have the arthritis, the test is correct 85% of the time. For individuals in this age group known to NOT have the arthritis, the test indicates arthritis (incorrectly!) 4% of the time. P( has arthritis | tests positive ) = ?

Arthritis 10% of this age group suffers from this arthritis. P(have arthritis) = 0.10 For individuals in this age group known to have the arthritis, the test is correct 85% of the time. P( tests positive | have arthritis ) = 0.85 For individuals in this age group known to NOT have the arthritis, the test indicates arthritis (incorrectly!) 4% of the time. P(tests positive | no arthritis ) = 0.04 P( have arthritis | tests positive ) = ?

0.85 positive Has arthritis negative 0.1 0.9 No arthritis 0.04 positive negative

The 3 Urns Three urns contain colored balls. Urn Red White Blue 1 3 4 1 2 1 2 3 3 4 3 2 An urn is selected at random and one ball is randomly selected from the urn. Given that the ball is red, what is the probability it came from urn #2 ?

The Lost Labels A large stockpile of cases of light bulbs, 100 bulbs to a box, have lost their labels. The boxes of bulbs come in 3 levels of quality: high, medium, and low. It’s known 50% of the boxes were high quality, 25% medium, and 25% low. Two bulbs will be tested from a box to check if they’re defective.

Lost Labels… The likelihood of finding defective bulbs is dependent on the bulb quality: Number of defects Low Medium High 0 .49 .64 .81 1 .42 .32 .18 2 .09 .04 .01 Given neither bulb is found to be defective, what is the probability the bulbs came from a box of high quality bulbs?