Conditional Probability and Independence

Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests. What is the probability that a randomly selected DWI suspect is given A test? A blood test or a breath test, but not both? Neither test?

Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests. What is the probability that a randomly selected DWI suspect is given A test? A blood test or a breath test, but not both? Neither test? Breath test Breath test Blood test Blood test 0.56 0.56 0.22 0.22 0.14 0.14 0.08 0.08

What is the probability that a randomly selected DWI suspect is given A test? P( breath test or blood test) = 0.56 + 0.22 + 0.14 = 0.92 Breath test Breath test Blood test Blood test 0.56 0.56 0.22 0.22 0.14 0.14 0.08

What is the probability that a randomly selected DWI suspect is given 2. A blood test or a breath test, but not both? P(blood or breath but not both) = 0.92 -.22 = 0.70 Breath test Breath test Blood test Blood test 0.56 0.56 0.22 0.22 0.14 0.14 0.08 0.08

What is the probability that a randomly selected DWI suspect is given 3. Neither test? P(neither test) = 1 – P(either test) = 1 – 0.92 = .08 Breath test Breath test Blood test Blood test 0.56 0.56 0.22 0.22 0.14 0.14 0.08 0.08

P(blood test and breath test) = 0.22 so not mutually exclusive Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests. Are giving a DWI suspect a blood test and a breath test mutually exclusive? 2. Are giving the two tests independent? P(blood test and breath test) = 0.22 so not mutually exclusive

Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath test, 36% a blood test, and 22% both tests. 2. Are giving the two tests independent? Breath test Yes No Total 0.22 0.36 0.78 1.00 Blood test

The two events are not independent! Are giving the two tests independent? Breath test Yes No Total 0.22 0.14 0.36 0.56 0.08 0.64 0.78 1.00 Blood test Does getting a breath test change the probability of getting a blood test? Does P(blood test) = P(blood | breath test)? P(blood test) = 0.36 P(blood | breath test) = 0.22/0.78 = 0.28 The two events are not independent!

Definition of Independent Events Two events E and F are independent if and only if P(F | E) = P(F) or P(E | F) = P(E)

EXAMPLE Illustrating Independent Events The probability a randomly selected murder victim is male is 0.7515. The probability a randomly selected murder victim is male given that they are less than 18 years old is 0.6751. Since P(male) = 0.7515 and P(male | < 18 years old) = 0.6751, the events “male” and “less than 18 years old” are not independent. In fact, knowing the victim is less than 18 years old decreases the probability that the victim is male.

Given a deck of cards, one card is drawn Given a deck of cards, one card is drawn. What is the probability that it is an ace or a red card? P(ace or red) = P(ace) + P(red) – P(ace and red) = + - =

Given a deck of cards, five cards are drawn without replacement Given a deck of cards, five cards are drawn without replacement. What is the probability they are all hearts? x x x x = 0.0005

I draw one card and look at it. I tell you it is red I draw one card and look at it. I tell you it is red. What is the probability it is a heart? P( heart | red) =

Are “red card” and “spade” mutually exclusive? Are they independent? Mutually exclusive events are always dependent! A red card can’t be a spade so they ARE mutually exclusive P(red) ? P(red|spade) .50 ≠ 0 So are NOT independent

Are “red card” and “ace” mutually exclusive? Are they independent? 2 aces are red cards so they are NOT mutually exclusive P(red) ? P(red|ace) .50 = .50 So they ARE independent

Are “face card” and “king” mutually exclusive? Are they independent? Kings are Face cards so they are NOT mutually exclusive P(face) ? P(face|king) 12/52 ≠ 1 So they are NOT independent

A company’s records indicate that on any given day about 1% of their day shift workers and 2% of their night shift workers will miss work. Sixty percent of the workers work the day shift. What percent of workers are absent on any given day? Is absenteeism independent of shift worked? 1.4% No, the probability depended on whether they worked the day (.01) or the night (.02) shift.

P(P|G) = .17/.64 = .266 No, 17% of homes have both. Real estate ads suggest that 64% of homes for sale have garages, 21% have swimming pools, and 17% have both features. If a home for sale has a garage, what’s the probability that it has a pool too? P(P|G) = .17/.64 = .266 Are having a garage and a pool mutually exclusive? No, 17% of homes have both. Are having a garage and a pool independent events? P(P) ? P(P|G) .21≠ .266 They are not independent.

Sample space = {(B, B), (B, G), (G, B), (G, G)} TWO-KID FAMILIES Consider the sample space of all families with two children: Sample space = {(B, B), (B, G), (G, B), (G, G)} Assume that male/female is equally likely.

Two Questions: 1. What is the probability of obtaining a family with two girls, given that the family has at least one girl? 2. What is the probability of obtaining a family with two girls, given that the older sibling is a girl?

Question 1: P(family with 2 girls | family has at least one girl) = P(family with two girls and family with at least one girl) P(family with at least one girl) = P(GG) P(GG or BG or GB) =

Question 2: P(family with 2 girls | family with older sibling a girl) = P(family with two girls and family with older sibling a girl) P(family with older sibling a girl) = P(GG) P(GG or GB) =

HUH????? I PROBABILITY!