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    5. AHL 4.13—Bayes theorem

    AHL 4.13—Bayes theorem

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    Question 1
    HLPaper 1

    A medical test for a certain disease has a 98% sensitivity and a 95% specificity. The prevalence of the disease in the population is 1%.

    1.

    Calculate the probability that a person selected at random from the population tests positive for the disease.

    [3]
    2.

    Given that a person tests positive, calculate the probability that they actually have the disease.

    [4]
    Question 2
    HLPaper 1

    A factory produces widgets using two machines, A and B. Machine A produces 60% of the widgets, and Machine B produces 40%. 2% of the widgets produced by Machine A are defective, while 5% of the widgets produced by Machine B are defective.

    1.

    Calculate the probability that a randomly selected widget is defective.

    [3]
    2.

    Given that a randomly selected widget is defective, calculate the probability that it was produced by Machine A.

    [4]
    Question 3
    HLPaper 2

    Occurrences XXX and YYY are such that P(X∪Y)=0.95{P}(X \cup Y) = 0.95P(X∪Y)=0.95, P(X∩Y)=0.6{P}(X \cap Y) = 0.6P(X∩Y)=0.6 and P(X∣Y)=0.75{P}(X|Y) = 0.75P(X∣Y)=0.75.

    1.

    Find P(Y){\text{P}}(Y)P(Y).

    [2]
    2.

    Find P(X)P(X)P(X).

    [2]
    3.

    Hence show that occurrences X′X'X′ and YYY are independent.

    [2]