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# Multiplicative Rule and Independent Events

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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}

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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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}

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}

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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