Chapter 7 Sets & Probability Section 7.5 Conditional Probability; Independent Events

Lie detectors are not admitted as evidence in courtroom trials due to the fact they are not 100% reliable. An experiment was conducted in which a group of suspects was instructed to lie or tell the truth to a set of questions, and a group of polygraph experts, along with the polygraph (lie detector), judged whether the suspect was telling the truth or not. The results are tabulated below. Suspects’ Answers Experts’ Judgment Truth Lie 93 11 7 89 Totals 100

Estimate the probability that: a.) there will be a miscarriage of justice. 18 / 200 = .09 b.) a suspect is a liar and gets away with the lie. 11 / 200 = .055 c.) the experts’ judgment is correct. 182 / 200 = .91 Suspects’ Answers Experts’ Judgment Truth Lie 93 11 7 89 Totals 100

Estimate the probability that: d.) a suspect is telling the truth 100 / 200 = .5 e.) a suspect is found to be honest 104 / 200 = .52 f.) a suspect who is telling the truth is found to be honest by the experts. 93 / 100 = .93 Suspects’ Answers Experts’ Judgment Truth Lie 93 11 7 89 Totals 100

Conditional Probability A probability problem in which the sample space is reduced by known, or given, information is called a conditional probability. In other words, a conditional probability exists when the sample space has been limited to only those outcomes that fulfill a certain condition.

Yes No Don’t Know Total Men 162 95 23 280 Women 256 45 19 320 418 140 In a newspaper poll concerning violence on television, 600 people were asked, “What is your opinion of the amount of violence on prime-time television – is there too much violence on television?” Their responses are indicated in the table below. Yes No Don’t Know Total Men 162 95 23 280 Women 256 45 19 320 418 140 42 600

Too Much Violence on Television? Yes No Don’t Know Total Men 162 95 23 280 Women 256 45 19 320 418 140 42 600 Use the table to find the probabilities below. P (Y) P (M) P (Y | M) P (M | Y) P (Y  M) P (M  Y)

A pair of dice is rolled. Find the probabilities of the given events. a.) The sum is 12 b.) The sum is 12, given that the sum is even c.) The sum is 12, given that the sum is odd d.) The sum is even, given that the sum is 12 e.) The sum is 4, given that the sum is less than 6 f.) The sum is less than 6, given that the sum is 4

A single die is rolled. Find the probabilities of the given events. a.) rolling a 5 b.) rolling a 5, given that the number rolled is odd c.) rolling an odd number, given that a 5 was rolled

A pair of dice is rolled. Find the probabilities of the given events. a.) sum is 10 b.) sum is 10, given the sum is even c.) sum is 7, given the sum is odd d.) sum is even, given the sum is 8

Product Rule of Probability The Product Rule gives a method for finding the probability that events E and F both occur.

Example Two cards are drawn without replacement from a standard deck of 52 cards. a.) Find the probability of getting a King followed by an Ace. b.) Find the probability of drawing a 7 and a Jack. c.) Find the probability of drawing two Aces.

The Nissota Automobile Company buys emergency flashers from two different manufacturers: one in Arkansas and one in Nevada. Thirty-nine percent of its turn-signal indicators are purchased from the Arkansas manufacturer, and the rest are purchased from the Nevada manufacturer. Two percent of the Arkansas turn-signal indicators are defective, and 1.7% of the Nevada indicators are defective. a.) What percent of the defective indicators are made in Arkansas? b.) What percent of the defective indicators are made in Nevada?

Manufacturer Indicator .02 Defective .39 Arkansas .98 Not Defective .61 . 017 Defective Nevada .983 Not Defective

Manufacturer Indicator P(D | A) .02 Defective P(A  D) = P(A) ∙ P(D | A) P(A) P(A  D) = .39 ∙ .02 = .0078 .39 Arkansas .98 Not Defective P(D | N) .61 . 017 Defective P(N  D) = P(N) ∙ P(D | N) P(N) P(N  D) = .61 ∙ .017 = .01037 Nevada .983 Not Defective a.) What percent of the defective indicators are made in Arkansas? P (A  D ) P(D) ____.0078____ .0078 + .01037 P ( A | D) = = ≈ .4293 ≈ 42.93%

Manufacturer Indicator .02 Defective .39 Arkansas .98 Not Defective .61 . 017 Defective Nevada .983 Not Defective b.) What percent of the defective indicators are made in Nevada? P ( N | D) = 1 - P ( A | D) = 1 - .4293 = .5707 = 57.07% (Using Complement Rule)

Two cards are dealt from a full deck of 52 Two cards are dealt from a full deck of 52. Find the probabilities of the given events. (Hint: Make a tree diagram, labeling each branch with the appropriate probabilities.) a.) The first card is a king. b.) Both cards are kings c.) The second card is a king, given that the first card was a king. d.) The second card is a king

A coin is flipped twice in succession A coin is flipped twice in succession. Find the probabilities of the given events. a.) Both tosses result in tails b.) The second toss is tails given the first toss is tails c.) The second toss results in tails d.) Getting tails on one toss and heads on the other

Independent Events Two events are independent if the probability of one event does not depend / affect the probability (or occurrence) of the other event. In other words, knowing F does not affect E’s probability.

Product Rule for Independent Events

Dependent Events Two events are dependent if the probability of one event does affect the probability (or likelihood of occurrence) of the other event. Two events E and F are dependent if P(E | F)  P(E) or P(F | E)  P(F)

Use your own personal experience or probabilities to determine whether the following events E and F are mutually exclusive and/or independent. a.) E is the event “being a doctor” and F is the event “being a woman”. b.) E is the event “it’s raining” and F is the event “it’s sunny”. c.) E is the event “being single” and F is the event “being married”. d.) E is the event “having naturally blond hair” and F is the event “having naturally black hair”. e.) If a die is rolled once, and E is the event “getting a 4” and F is the event “getting an odd number”. f.) If a die is rolled once, and E is the event “getting a 4” and F is the event “getting an even number”.

A skateboard manufacturer buys 23% of its ball bearing from a supplier in Akron, 38% from one in Atlanta, and the rest from a supplier in Los Angeles. Of the ball bearings from Akron, 4% are defective; 6.5% of those from Atlanta are defective; and 8.1% of those from Los Angeles are defective. a.) Find the probability that a ball bearing is defective. b.) Are the events “defective” and “from the Los Angeles supplier” independent? Show mathematical justification. c.) Are the events “defective” and “from the Atlanta supplier” independent? Show mathematical justification. d.) What conclusion can you draw?