Answer:
a
[tex]P(X = 4 ) = 0.1876[/tex]
b
[tex]P(X \le 4) = 0.8358[/tex]
Step-by-step explanation:
From the question we are told that
The proportion that has outstanding balance is p = 0.20
The sample size is n = 15
Given that the properties of the binomial distribution apply, for a randomly selected number(X) of credit card
[tex]X \ \ ~ Bin (n , p )[/tex]
Generally the probability of finding 4 customers in a sample of 15 who have "maxed out" their credit cards is mathematically represented as
[tex]P(X = 4 ) = ^nC_4 * p^4 * (1 - p)^{n-4}[/tex]
=> [tex]P(X = 4 ) = ^{15}C_4 * (0.20)^4 * (1 - 0.20)^{15-4}[/tex]
Here C stand for combination
=> [tex]P(X = 4 ) = 0.1876[/tex]
Generally the probability that 4 or fewer customers in the sample will have balances at the limit of the credit card is mathematically represented as
[tex]P(X \le 4) = [ ^{15}C_0 * (0.20)^0 * (1 - 0.20)^{15-0}]+[ ^{15}C_1 * (0.20)^1 * (1 - 0.20)^{15-1}]+\cdots+[ ^{15}C_4 * (0.20)^4 * (1 - 0.20)^{15-4}][/tex]
=> [tex]P(X \le 4) = 0.8358[/tex]
