Forty percent of households say they would feel secure if they had $50,000 in savings. you randomly select 8 households and ask them if they

Question

Forty percent of households say they would feel secure if they had $50,000 in savings. you randomly select 8 households and ask them if they would feel secure if they had $50,000 in savings. find the probability that the number that say they would feel secure is (a) exactly five, (b) more than five, and (c) at most five.

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Madelyn 1 week 2021-09-15T19:20:14+00:00 1 Answer 0

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    2021-09-15T19:21:40+00:00

    Answer:

    Let X be the event of feeling secure after saving $50,000,

    Given,

    The probability of feeling secure after saving $50,000, p = 40 % = 0.4,

    So, the probability of not  feeling secure after saving $50,000, q = 1 – p = 0.6,

    Since, the binomial distribution formula,

    P(x=r)=^nC_r p^r q^{n-r}

    Where, ^nC_r=\frac{n!}{r!(n-r)!}

    If 8 households choose randomly,

    That is, n = 8

    (a) the probability of the number that say they would feel secure is exactly 5

    P(X=5)=^8C_5 (0.4)^5 (0.6)^{8-5}

    =56(0.4)^5 (0.6)^3

    =0.12386304

    (b) the probability of the number that say they would feel secure is more than five

    P(X>5) = P(X=6)+ P(X=7) + P(X=8)

    =^8C_6 (0.4)^6 (0.6)^{8-6}+^8C_7 (0.4)^7 (0.6)^{8-7}+^8C_8 (0.4)^8 (0.6)^{8-8}

    =28(0.4)^6 (0.6)^2 +8(0.4)^7(0.6)+(0.4)^8

    =0.04980736

    (c) the probability of the number that say they would feel secure is at most five

    P(X\leq 5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)

    =^8C_0 (0.4)^0(0.6)^{8-0}+^8C_1(0.4)^1(0.6)^{8-1}+^8C_2 (0.4)^2 (0.6)^{8-2}+8C_3 (0.4)^3 (0.6)^{8-3}+8C_4 (0.4)^4 (0.6)^{8-4}+8C_5(0.4)^5 (0.6)^{8-5}

    =0.6^8+8(0.4)(0.6)^7+28(0.4)^2(0.6)^6+56(0.4)^3(0.6)^5+70(0.4)^4(0.6)^4+56(0.4)^5(0.6)^3

    =0.95019264

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