Question

according to a recent survey, 31 percent of the residents of a certain state who are age 25 years or older have a bachelor’s degree. a random sample of 50 residents of the state, age 25 years or older, will be selected. let the random variable ( b ) represent the number in the sample who have a bachelor’s degree. what is the probability that ( b ) will equal 40 ?

a ( dbinom{50}{40}(0.31)^{40}(0.69)^{10} )

b ( dbinom{50}{40}(0.69)^{40}(0.31)^{10} )

c ( dbinom{40}{10}(0.31)^{40}(0.69)^{10} )

Explanation:

Step1: Identify distribution type

This is a binomial probability problem, where each trial (resident) is independent, with two outcomes: has a bachelor's degree (success) or not (failure).

Step2: Define binomial parameters

Number of trials $n=50$, probability of success $p=0.31$, number of successes $k=40$, probability of failure $q=1-p=0.69$.

Step3: Apply binomial formula

The binomial probability formula is $P(X=k)=\binom{n}{k}p^kq^{n-k}$. Substitute the values:
$$P(B=40)=\binom{50}{40}(0.31)^{40}(0.69)^{50-40}=\binom{50}{40}(0.31)^{40}(0.69)^{10}$$

Answer:

A. $\dbinom{50}{40}(0.31)^{40}(0.69)^{10}$