# Multiplicative Rule and Independent Events

48. For two events, A and B, P(A) = 0.4, P(B) = 0.2, and P(A/B)=0.6

a. find P(A and B)

b. find P(B/A)

50. An experiment results in one of three mutually exclusive events, A,B, or C. It is known that P(A) =.30,

P(B) = .55 and P (C) = .15. Find each of the following probabilities:

a. P(A U B)

b. P(A and C)

c. P(A/B)

d. P(B U C)

e. are B and C independent events? Explain

52. Two fair coins are tossed, and the following events are defined:

A: {observe at least one head}

B: {observe exactly one head}

a. Draw a Venn diagram for the experiment, showing events A and B. Assign probabilities to the sample

points.

b. Find P(A), P(B), and P(A upside down U B)

c. Use the formula for conditional probability to find P(A/B) and P(B/A). Verify your answer by inspecting the

Venn diagram and using the concept of reduced sample spaces.

54. Two fair dice are tossed and the following events are defined:

A: {sum of the numbers showing is odd}

B: {sum of the numbers showing is 9, 11, 12}

Are events A and B independent? Why

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

48. For two events, A and B, P(A) = 0.4, P(B) = 0.2, and P(A/B)=0.6

a. find P(A and B)

Solution: P (A and B) =P(B)*P(A|B)=0.2*0.6=0.12

b. find P(B/A)

Solution: P(B|A)= P (A and B)/P(A)=0.12/0.4=0.3

50. An experiment results in one of three mutually exclusive events, A,B, or C. It is known that P(A) =.30,

P(B) = .55 and P (C) = .15. Find each of the following probabilities:

a. P(AUB)

=P(A)+P(B) =0.30+0.55=0.85

b. P(A and C)

=0

c. P(A/B)

=0

d. P(B U C)

=P(B)+P(C)=0.55+0.15=0.70

e. are B and C independent events? Explain

As B and C are mutually exclusive events, P(B and C)=0. So, P(B and C) P(B)*P(C)=0.55*0.15=0.0825.

Hence, B and C are NOT independent

52. Two fair coins are tossed, and the following events are defined:

A: {observe at least one head}

B: {observe exactly one head}

a. Draw a Venn diagram for the experiment, showing events A and B. Assign probabilities to the sample

points => See attachment for diagram.

A={HH,HT,TH}

B={HT,TH}

Where H denotes "head" and T denotes "tail"

P(HH)=P(HT)=P(TH)=P(TT)=1/4

b. Find P(A), P(B), and P(A upside down U B)

P(A)=3/4 and P(B)=2/4=1/2

P (A B)=P(B)=1/2

c. Use the formula for conditional probability to find P(A/B) and P(B/A). Verify your answer by inspecting the

Venn diagram and using the concept of reduced sample spaces.

P(A|B)= P (A and B)/P(B)= (1/2)/(1/2)=1

P(B|A)= P (A and B)/P(A)= (1/2)/(3/4)=2/3

For the conditional probability of B given A occurs, we look at the reduced sample space {HH, HT,TH}. As B={HT, TH}, P(B|A)=2/3.

For the conditional probability of A given B occurs, we look at the reduced sample space {HT,TH}. As A={HH, HT, TH}, P(A|B)=2/2=1.

54. Two fair dice are tossed and the following events are defined:

A: {sum of the numbers showing is odd}

B: {sum of the numbers showing is 9, 11, 12}

Are events A and B independent? Why

Solution. As A and B = {sum of the numbers showing is 9, 11}, P(A and B) = 6/36 = 1/6.

However, P(A) = 1/2 and P(B) = 7/36. So,

P(A)P(B) = 1.2*7/36 = 7/72 does not equal P(A and B)

So A and B are not independent.

So, A and B are not independent.

52. Two fair coins are tossed, and the following events are defined:

A: {observe at least one head}

B: {observe exactly one head}

a. Draw a Venn diagram for the experiment, showing events A and B. Assign probabilities to the sample points.

A={HH,HT,TH}

B={HT,TH}

Where H denotes "head" and T denotes "tail"

P(HH)=P(HT)=P(TH)=P(TT)=1/4

b. Find P(A), P(B), and P(A upside down U B)

P(A)=3/4 and P(B)=2/4=1/2

P (A B)=P(B)=1/2

c. Use the formula for conditional probability to find P(A/B) and P(B/A). Verify your answer by inspecting the Venn diagram and using the concept of reduced sample spaces.

Solution.

P(A|B)= P (A B)/P(B)= (1/2)/(1/2)=1

P(B|A)= P (A B)/P(A)= (1/2)/(3/4)=2/3

For the conditional probability of B given A occurs, we look at the reduced sample space {HH, HT,TH}. As B={HT, TH}, P(B|A)=2/3.

For the conditional probability of A given B occurs, we look at the reduced sample space {HT,TH}. As A={HH, HT, TH}, P(A|B)=2/2=1.

54. Two fair dice are tossed and the following events are defined:

A: {sum of the numbers showing is odd}

B: {sum of the numbers showing is 9, 11, 12}

Are events A and B independent? Why

Solution. As {sum of the numbers showing is 9, 11}, .

However, and . So,

So, A and B are not independent.

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