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.
$\begin{aligned}
P(k \text { successes }) & ={ }_n C_k p^k(1-p)^{n-k} \\
{ }_n C_k & =\frac{n!}{(n-k)|\cdot k|}
\end{aligned}$
A. 0.024
B. 0.031
C. 0.132
D. 0.309
Answer (1)
Use the binomial probability formula: P ( k successes ) = n C k p k ( 1 − p ) n − k .
Calculate the binomial coefficient: 5 C 2 = 10 .
Calculate the probability: P ( 2 successes ) = 10 ⋅ ( 0.3 ) 2 ⋅ ( 0.7 ) 3 = 0.3087 .
Round to the nearest thousandth: 0.309 .
Explanation
Understand the problem and provided data We are given that 30% of voters support a ballot initiative, meaning the probability of a single voter supporting the initiative is p = 0.3 . Hans surveys 5 voters, and we want to find the probability that exactly 2 of them support the initiative. This is a binomial probability problem.
State the binomial probability formula The binomial probability formula is given by: P ( k successes ) = n C k p k ( 1 − p ) n − k where n is the number of trials, k is the number of successes, and p is the probability of success on a single trial. In this case, n = 5 , k = 2 , and p = 0.3 .
Calculate the binomial coefficient First, we calculate the binomial coefficient n C k , which represents the number of ways to choose k successes from n trials: 5 C 2 = ( 5 − 2 )! ⋅ 2 ! 5 ! = 3 ! ⋅ 2 ! 5 ! = ( 3 ⋅ 2 ⋅ 1 ) ( 2 ⋅ 1 ) 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 2 ⋅ 1 5 ⋅ 4 = 10 So, there are 10 ways to choose 2 voters out of 5.
Calculate p^k Next, we calculate p k , which is the probability of k successes: p k = ( 0.3 ) 2 = 0.09
Calculate (1-p)^(n-k) Then, we calculate ( 1 − p ) n − k , which is the probability of n − k failures: ( 1 − p ) n − k = ( 1 − 0.3 ) 5 − 2 = ( 0.7 ) 3 = 0.343
Calculate the final probability and round Now, we plug these values into the binomial probability formula: P ( 2 successes ) = 5 C 2 ⋅ p 2 ⋅ ( 1 − p ) 3 = 10 ⋅ 0.09 ⋅ 0.343 = 0.3087 Rounding to the nearest thousandth, we get 0.309 .
State the final answer Therefore, the probability that exactly 2 voters will be in favor of the ballot initiative is approximately 0.309 .
Examples
Consider a quality control scenario where a company produces items, and 30% of them are defective. If you randomly select 5 items, the probability of finding exactly 2 defective items can be calculated using the same binomial probability formula. This helps the company understand the likelihood of different defect rates in their samples, aiding in process improvement and quality assurance.
