Forty-four percent of customers who visit a department store make a purchase. What is the probability that in a random sample of 9 customers

Question

Forty-four percent of customers who visit a department store make a purchase. What is the probability that in a random sample of 9 customers who will visit this department store, exactly 6 will make a purchase?

A. .1070

B. .1752

C. .8930

D. .0033

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Aubrey 2 weeks 2021-09-26T16:32:30+00:00 2 Answers 0

Answers ( )

    0
    2021-09-26T16:34:13+00:00

    Answer:

    A. .1070

    Step-by-step explanation:

    For each customer, there are only two possible outcomes. Either they make a purchase, or they do not. The probability of a customer making a purchase is independent from other customers. So we use the binomial probability distribution to solve this question.

    Binomial probability distribution

    The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

    P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

    In which C_{n,x} is the number of different combinations of x objects from a set of n elements, given by the following formula.

    C_{n,x} = \frac{n!}{x!(n-x)!}

    And p is the probability of X happening.

    Forty-four percent of customers who visit a department store make a purchase.

    This means that p = 0.44

    What is the probability that in a random sample of 9 customers who will visit this department store, exactly 6 will make a purchase?

    This is P(X = 6) when n = 9. So

    P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

    P(X = 6) = C_{9,6}.(0.44)^{6}.(0.56)^{3} = 0.1070

    So the correct answer is:

    A. .1070

    0
    2021-09-26T16:34:20+00:00

    Answer: A. .1070

    Step-by-step explanation:

    We would apply the formula for binomial distribution which is expressed as

    P(x = r) = nCr × p^r × q^(n – r)

    Where

    x represent the number of successes.

    p represents the probability of success.

    q = (1 – r) represents the probability of failure.

    n represents the number of customers sampled.

    From the information given,

    p = 44% = 44/100 = 0.44

    q = 1 – p = 1 – 0.44

    q = 0.56

    n = 9

    x = r = 6

    Therefore,

    P(x = 6) = 9C6 × 0.44^6 × 0.56^(9 – 6)

    P(x = 6) = 84 × 0.0073 × 0.175616

    P(x = 6) = 0.107

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