Question

according to a poll, 30% of voters support a ballot initiative. hans randomly surveys 5 voters. what is the probability that exactly 2 voters will be in favor of the ballot initiative? round the answer to the nearest thousandth.

$p(k\text{ successes}) = _nc_kp^k(1 - p)^{n - k}$

$_nc_k=\frac{n!}{(n - k)!k!}$

0.024
0.031
0.132
0.309

Explanation:

Step1: Identify values of n, k, p

$n = 5$ (number of voters surveyed), $k = 2$ (number of voters in - favor), $p=0.3$ (probability of a voter in - favor)

Step2: Calculate $_{n}C_{k}$

$_{n}C_{k}=\frac{n!}{(n - k)!k!}=\frac{5!}{(5 - 2)!2!}=\frac{5!}{3!2!}=\frac{5\times4\times3!}{3!\times2\times1}=10$

Step3: Calculate $(1 - p)^{n - k}$

$1-p = 1 - 0.3=0.7$, $n - k=5 - 2 = 3$, so $(1 - p)^{n - k}=0.7^{3}=0.343$

Step4: Calculate $p^{k}$

$p^{k}=0.3^{2}=0.09$

Step5: Calculate probability $P(k)$

$P(2)=_{n}C_{k}p^{k}(1 - p)^{n - k}=10\times0.09\times0.343 = 0.3087\approx0.309$

Answer:

0.309