SECTION 6.3

GENERAL PROBABILITY RULES

General addition ruleP(A or B) = P(A) + P(B) – P(A and B)

Addition rule for disjoint eventsP(one or more of A, B, C) = P(A) + P(B) + P(C)

Multiplication rule for independent eventsP(A and B) = P(A)P(B)

Review of Previous Rules

Conditional probability – the probability of one event under the condition that we know another event.

The “|” can be interpreted as “given the information that”

General addition rule P(A or B) = P(A) + P(B) – P(A and B) P(A U B) = P(A) + P(B) – P(A ∩ B)

General multiplication rule P(A and B) = P(A)P(B given A) P(A ∩ B) = P(A)P(B|A)

When P(A) > 0, Testing for independence:

Two events A and B are independent if P(B|A) = P(B)

TESTING FOR INDEPENDENCE

Two events A and B are independent if P(B|A) = P(B)

Think back to your last quiz. When rolling a die and then flipping a coin, let event A be getting a 1 or 2 on the roll of the die. Let event B be getting an even number on the die. Are A and B independent?

P(B|A) = (2/12)/(4/12) = ½

P(B) = 6/12 = ½

Therefore, A and B are independent.

A

A B

BLUE REPRESENTS DESIGNATED AREA

AC

A B

BLUE REPRESENTS DESIGNATED AREA

B

A B

BLUE REPRESENTS DESIGNATED AREA

BC

A B

BLUE REPRESENTS DESIGNATED AREA

A∩B

A B

BLUE REPRESENTS DESIGNATED AREA

(A∩B)C

A B

BLUE REPRESENTS DESIGNATED AREA

AUB

A B

BLUE REPRESENTS DESIGNATED AREA

(AUB)C

A B

BLUE REPRESENTS DESIGNATED AREA

Taste In Music Musical styles other than rock and pop are

becoming more popular. A survey of college students find that 40% like country music, 30% like gospel music, and 10% like both. What is the conditional probability that a student

likes gospel music if we know that he/she likes country music?

C G

10%30% 20%

Conditional Probability

P(G|C) = 0.1/0.4=0.25

Taste In Music (cont.) Musical styles other than rock and pop are

becoming more popular. A survey of college students find that 40% like country music, 30% like gospel music, and 10% like both. What is the conditional probability that a student

who does not like country music likes gospel music?

C G

10%30% 20%

Conditional Probability

P(G|CC) = 0.2/0.6=1/3

Venn Diagram PracticeRESTAURANT

11

23

13

15

7

10

129

T C

Q

RESTAURANT

1. (CUTUQ)C P(CUTUQ)C = 23/100 2. (C∩T∩Q) P (C∩T∩Q) = 11/100 3. (CUQ∩TC) P(CUQ∩TC) = 29/100 4. (Q) P(Q) = 41/100 5. (T∩Q)U(Q∩C)U(C∩T)

P(T∩Q)U(Q∩C)U(C∩T) = 46/100 6. (T∩Q∩CC) P(T∩Q∩CC) =

13/100 7. (T∩C) P (T∩C) = 26/100

Venn Diagram PracticeCARTOONS

T A

P 22

73

14319

23 17

11

Venn Diagram PracticeCONCERT

P D

G

D

G 18

1013

11

15

35

2721

Venn Diagram PracticeSTAR TREK

T D

V

73

31

22

14

1723

9

11

Venn Diagram PracticeMYTHOLOGY

L H

R25

37

5

6

24

1812

Venn Diagram PracticePOLLUTANTS

C P

S177

101

72

137

15228

122

211

Venn Diagram PracticeTENNIS

S B

F20

3010

40

5

5235

8

Venn Diagram PracticeTENNIS TOURNAMENTS

U W

A

30

10

1520

30

5

5040

Venn Diagram PracticeLANGUAGES

F G

S

416

820

27

12

92

29

Nobel Prize WinnersCOUNTRY PHYSIC

SCHEMISTR

YPhys/Med

Total

United States 74 51 90 215

United Kingdom

21 27 28 76

Germany 19 29 15 63

France 11 7 7 25

Soviet Union 9 1 2 12

Japan 4 4 0 8

TOTAL 138 119 142 399If a laureate is selected at random, what is the probability that:

a) his or her award is in chemistry?

b) the award was won by someone working in the US?

c) the awardee was working in the US, given the award was for phys./med?

d) the award was for phys./med., given that the awardee was working in the US?

119/399 ≈ 0.2982

215/399 ≈ 0.5388

90/142 ≈ 0.6338

90/215 ≈ 0.4186