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10 July, 02:04

Suppose the probability of contracting a certain disease is 1 in 27,124 for a new case in a given year. Approximate the probability that in a town of 5,756 people there will be at least one new case of the disease next year.

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  1. 10 July, 02:07
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    The probability that there is at least one new case of the disease is approximately 0.19121

    Step-by-step explanation:

    Given that the probability of contracting a disease is p = 1/27124.

    In an experiment n = 5756, we want to find the probability that there will be at least one new case of the disease. That is P (X ≥ 1).

    If x is approximated Binomial (n, p)

    Then

    P (X = x) = (nCx) (p^x) q^ (n - x)

    Where q = 1 - p

    Here, q = 1 - (1/27124) = 27123/27124

    And nCx, read as "n combination x"

    is given as n! / (n - x) ! x!

    Also note that

    P (X ≥ a) = 1 - P (X < a)

    So, it is sufficient to find P (X < 1) for this problem.

    P (X < 1) = P (X = 0)

    = (5756C0) (1/27124) ^0 (27123/27124) ^ (5756 - 0)

    = 1 * 1 * 0.80879

    ≈ 0.80879

    Now,

    P (X ≥ 1) = 1 - 0.80879 = 0.19121
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