A study indicates that 62% of students have have a laptop. You randomly sample 8 students. Find the probability that between 4 and 6 (includ

Question

A study indicates that 62% of students have have a laptop. You randomly sample 8 students. Find the probability that between 4 and 6 (including endpoints) have a laptop.

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Gabriella 1 month 2021-10-10T02:45:27+00:00 1 Answer 0 views 0

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    2021-10-10T02:47:20+00:00

    Answer:

    72.69% probability that between 4 and 6 (including endpoints) have a laptop.

    Step-by-step explanation:

    For each student, there are only two possible outcomes. Either they have a laptop, or they do not. The probability of a student having a laptop is independent from other students. 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.

    A study indicates that 62% of students have have a laptop.

    This means that n = 0.62

    You randomly sample 8 students.

    This means that n = 8

    Find the probability that between 4 and 6 (including endpoints) have a laptop.

    P(4 \leq X \leq 6) = P(X = 4) + P(X = 5) + P(X = 6)

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

    P(X = 4) = C_{8,4}.(0.62)^{4}.(0.38)^{4} = 0.2157

    P(X = 5) = C_{8,5}.(0.62)^{5}.(0.38)^{3} = 0.2815

    P(X = 6) = C_{8,6}.(0.62)^{6}.(0.38)^{2} = 0.2297

    P(4 \leq X \leq 6) = P(X = 4) + P(X = 5) + P(X = 6) = 0.2157 + 0.2815 + 0.2297 = 0.7269

    72.69% probability that between 4 and 6 (including endpoints) have a laptop.

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