Question

Forty-five 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.

Answer 1

0.172 ; 0.0884 ; 0.9115

Explanation:

Proportion or those who feel secure, p = 0.45

Sample size, n = 8

Using the binomial distribution formula :

P(x =x) = nCx * p^x * (1 - p)^(n - x)

A.) p(x = 5)

P(x =5) = 8C5 * 0.45^5 * 0.55^3

P(x = 5) = 56 * 0.0184528125 * 0.166375

P(x = 5) = 0.1719248540625

P(x = 5) = 0.172

B.) P(x > 5)

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

P(x > 5) = 0.0703 + 0.0164 + 0.0017

P(x > 5) = 0.0884

C.) P( ≤ 5) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5)

Using the binomial probability calculator to obtain a direct solution :

P( ≤ 5) = 0.9115