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

1.A test?2.A blood test or a breath test,

but not both?3.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

1. A test?2. A blood test or a breath test, but not both?3. Neither test?

Breath testBlood test

0.220.560.14

0.08

Breath testBlood test

0.220.560.14

0.08

What is the probability that a randomly selected DWI suspect is given

1.A test?

Breath testBlood test

0.220.560.14

Breath testBlood test

0.220.560.14

0.08

P( breath test or blood test) = 0.56 + 0.22 + 0.14 = 0.92

What is the probability that a randomly selected DWI suspect is given

2. A blood test or a breath test, but not both?

Breath testBlood test

0.220.560.14

0.08

Breath testBlood test

0.220.560.14

0.08

P(blood or breath but not both) = 0.92 -.22 = 0.70

What is the probability that a randomly selected DWI suspect is given

3. Neither test?

Breath testBlood test

0.220.560.14

0.08

Breath testBlood test

0.220.560.14

0.08

P(neither test) = 1 – P(either test) = 1 – 0.92 = .08

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.

1.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?

Yes No Total

Yes 0.22 0.36

No

Total 0.78 1.00

Breath test

Blo

od

test

Yes No Total

Yes 0.22 0.14 0.36No 0.56 0.08 0.64

Total 0.78 0.22 1.00

Breath test

Blo

od

test

Are giving the two tests independent?

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. 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) =

52

4

52

26

52

2

52

28+ - =

Given a deck of cards, five

cards are drawn

without replacement. What is the probability they are all

hearts?

52

13

51

12

50

11x x = 0.0005

49

1048

9x x

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) =

13P(heart and red) 152

26P(red) 252

Are “red card” and “spade” mutually

exclusive? Are they

independent?A red card can’t be a spade so they ARE mutually exclusive

P(red) ? P(red|spade).50 0

So are NOT independent

Mutually exclusive

events are always

dependent!

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.

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?

Are having a garage and a pool mutually exclusive?

Are having a garage and a pool independent events?

P(P|G) = .17/.64 = .266

No, 17% of homes have both.

P(P) ? P(P|G) .21≠ .266 They are not independent.

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)

= 3333.75.

25.

4/3

4/1

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)

= 50.50.

25.

2/1

4/1

HUH?????I

PROBABILITY!