Question

fifty - nine percent of u.s. adults think that civil rights for black americans have improved during their lifetime. you randomly select nine u.s. adults. find the probability that the number who think that civil rights for black americans have improved during their lifetime is (a) exactly two and (b) exactly six. (a) the the probability that the number who think that civil rights for black americans have improved during their lifetime is exactly two is 0.024. (round to three decimal places as needed.) (b) the the probability that the number who think that civil rights for black americans have improved during their lifetime is exactly six is 0.241. (round to three decimal places as needed.)

Explanation:

Step1: Identify the binomial - probability formula

The binomial - probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Determine the values of $n$, $p$, and $1 - p$

Given that $n = 9$ (the number of U.S. adults selected), $p=0.59$ (the probability that a U.S. adult thinks civil rights for Black Americans have improved), and $1 - p = 1-0.59 = 0.41$.

Step3: Calculate the probability for part (a) ($k = 2$)

First, calculate the combination $C(9,2)=\frac{9!}{2!(9 - 2)!}=\frac{9\times8}{2\times1}=36$. Then, $P(X = 2)=C(9,2)\times(0.59)^{2}\times(0.41)^{9 - 2}=36\times0.59^{2}\times0.41^{7}\approx36\times0.3481\times0.001807\approx0.024$.

Step4: Calculate the probability for part (b) ($k = 6$)

Calculate the combination $C(9,6)=C(9,3)=\frac{9!}{6!(9 - 6)!}=\frac{9\times8\times7}{3\times2\times1}=84$. Then, $P(X = 6)=C(9,6)\times(0.59)^{6}\times(0.41)^{9 - 6}=84\times0.59^{6}\times0.41^{3}=84\times0.0422\times0.068921\approx0.241$.

Answer:

(a) 0.024
(b) 0.241