Question

Thirty 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

Answer: a) 0.04667544

b) 0.01129221

c) 0.98870779

Explanation:

Binomial probability formula :-

$P(x)=^nC_xp^x(1-p)^x$, where P(x) is the probability of getting success in x trials, n is the total number of trials and p is the probability of getting success in each trial.

Given : The probability of households say they would feel secure if they had​ $50,000 in savings = 0.30

Total number of households selected = 8

a) The probability that the number that say they would feel secure is exactly​ five will be :-

$P(5)=^8C_{5}(0.30)^5(0.70)^3\\\\=56(0.3)^5(0.7)^3=0.04667544$

b) The probability that the number that say they would feel secure is more than​ five :-

$P(x>5)=P(6)+P(7)+P(8)\\\\=^8C_{6}(0.30)^6(0.70)^2+^8C_{7}(0.30)^7(0.70)^1+^8C_{0}(0.30)^8(0.70)^0\\\\=28(0.30)^6(0.70)^2+8(0.30)^7(0.70)^1+(0.30)^8\\\\=0.01129221$

c) The probability that the number that say they would feel secure is at most five :-

$P(x\leq5)=1-P(x>5)\\\\=1-0.01129221=0.98870779$