Question

In Exercises 1–5, assume that 74% of randomly selected adults have a credit card (basedon results from an AARP Bulletin survey). Assume that a group of five adults is randomly selected.

Find the probability that exactly three of the five adults have credit cards.

Short answer

The probability that exactly three of the five adults have credit cards is 0.274.

Step-by-step solution

Given information

The number of randomly selected adults are $n=5$.

The probability of randomly selected adults that have a credit card is $p=0.74$.

Compute the probability that exactly three of the five adults have credit cards

Let x represents the number of adults who have credit cards.

In the given scenario, the variable x will follow the binomial distribution.

The probability mass function of the binomial distribution is given as,

$P\left(x\right)={}_n{C}_x{p}^x{q}^{n-x}$

The probability that exactly three of the five adults have credit cards is computed as,

$$\begin{gathered} P\left(3\right)={}^5{C}_3{\left(0.74\right)}^3{\left(1-0.74\right)}^{5-3} \\ =\frac{5!}{3!\left(5-3\right)!}{\left(0.74\right)}^3{\left(0.26\right)}^2 \\ =0.274 \end{gathered}$$

Thus, the probability that exactly three of the five adults have credit cards is 0.274.