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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}
    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

    © BrainMass Inc. brainmass.com December 24, 2021, 10:44 pm ad1c9bdddf
    https://brainmass.com/statistics/probability/multiplicative-rule-independent-events-504189

    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.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 10:44 pm ad1c9bdddf>
    https://brainmass.com/statistics/probability/multiplicative-rule-independent-events-504189

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