To find the probability that exactly 2 out of 5 voters support the ballot initiative, given that 30% of voters support it, we will use the binomial probability formula.
The binomial probability formula is:
[tex]\[
P(k \text{ successes}) = {}_n C_k \cdot p^k \cdot (1 - p)^{n - k}
\][/tex]
Where:
- [tex]\(n\)[/tex] is the number of trials (voters being surveyed), which is 5 in this case.
- [tex]\(k\)[/tex] is the number of successes (voters supporting the initiative), which is 2 here.
- [tex]\(p\)[/tex] is the probability of success on an individual trial, given as 0.3 (30%).
- [tex]\({}_n C_k\)[/tex] is the binomial coefficient, also known as "n choose k", which calculates the number of ways to choose [tex]\(k\)[/tex] successes out of [tex]\(n\)[/tex] trials.
First, we calculate the binomial coefficient [tex]\({}_n C_k\)[/tex]:
[tex]\[
{}_5 C_2 = \frac{5!}{(5-2)! \cdot 2!} = \frac{5!}{3! \cdot 2!} = \frac{5 \cdot 4 \cdot 3!}{3! \cdot 2 \cdot 1} = \frac{5 \cdot 4}{2 \cdot 1} = 10
\][/tex]
Now we use the binomial probability formula:
[tex]\[
P(2 \text{ successes}) = 10 \cdot (0.3)^2 \cdot (1 - 0.3)^{5 - 2}
\][/tex]
Calculating each component:
[tex]\[
(0.3)^2 = 0.09
\][/tex]
[tex]\[
(1 - 0.3) = 0.7
\][/tex]
[tex]\[
0.7^{3} = 0.343
\][/tex]
Putting it all together:
[tex]\[
P(2 \text{ successes}) = 10 \cdot 0.09 \cdot 0.343
\][/tex]
[tex]\[
= 10 \cdot 0.03087
\][/tex]
[tex]\[
= 0.3087
\][/tex]
Rounding to the nearest thousandth, the probability that exactly 2 out of 5 voters support the ballot initiative is:
[tex]\[
0.309
\][/tex]
Therefore, the answer is:
[tex]\[
0.309
\][/tex]
