Answer :
To determine the probability that exactly 2 out of 5 voters will support the ballot initiative when [tex]\(30\%\)[/tex] of the voters support it, we will use the binomial probability formula. This calculates the probability of getting a fixed number of successes [tex]\(k\)[/tex] in [tex]\(n\)[/tex] independent Bernoulli trials, where each trial has a success probability [tex]\(p\)[/tex].
Given:
- [tex]\(p = 0.30\)[/tex]: Probability of success on a single trial (voter supports the initiative)
- [tex]\(n = 5\)[/tex]: Total number of trials (voters surveyed)
- [tex]\(k = 2\)[/tex]: Number of successes of interest (voters who support the initiative)
The binomial probability formula is:
[tex]\[ P(k \text{ successes}) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k} \][/tex]
where [tex]\(\binom{n}{k}\)[/tex] (the binomial coefficient) is calculated as:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \][/tex]
Let’s break down the calculations step by step:
1. Calculate the binomial coefficient [tex]\(\binom{n}{k}\)[/tex]:
[tex]\[ \binom{5}{2} = \frac{5!}{2!(5 - 2)!} = \frac{5!}{2!3!} \][/tex]
[tex]\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \][/tex]
[tex]\[ 2! = 2 \times 1 = 2 \][/tex]
[tex]\[ 3! = 3 \times 2 \times 1 = 6 \][/tex]
So,
[tex]\[ \binom{5}{2} = \frac{120}{2 \times 6} = \frac{120}{12} = 10 \][/tex]
2. Calculate the probability of exactly 2 successes ([tex]\(P(2)\)[/tex]):
[tex]\[ P(2 \text{ successes}) = \binom{5}{2} \cdot p^2 \cdot (1 - p)^{5 - 2} \][/tex]
Substitute [tex]\( \binom{5}{2} = 10 \)[/tex], [tex]\( p = 0.30 \)[/tex], and [tex]\( 1 - p = 0.70 \)[/tex]:
[tex]\[ P(2 \text{ successes}) = 10 \cdot (0.30)^2 \cdot (0.70)^3 \][/tex]
3. Calculate the individual terms:
[tex]\[ (0.30)^2 = 0.09 \][/tex]
[tex]\[ (0.70)^3 = 0.343 \][/tex]
4. Combine them:
[tex]\[ P(2 \text{ successes}) = 10 \cdot 0.09 \cdot 0.343 = 0.3087 \][/tex]
The probability that exactly 2 out of 5 voters will support the ballot initiative is approximately [tex]\(0.3087\)[/tex].
Finally, rounding this to the nearest thousandth gives:
[tex]\[ \boxed{0.309} \][/tex]
Thus, the correct answer from the given choices is:
[tex]\[ \boxed{0.309} \][/tex]
Given:
- [tex]\(p = 0.30\)[/tex]: Probability of success on a single trial (voter supports the initiative)
- [tex]\(n = 5\)[/tex]: Total number of trials (voters surveyed)
- [tex]\(k = 2\)[/tex]: Number of successes of interest (voters who support the initiative)
The binomial probability formula is:
[tex]\[ P(k \text{ successes}) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k} \][/tex]
where [tex]\(\binom{n}{k}\)[/tex] (the binomial coefficient) is calculated as:
[tex]\[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \][/tex]
Let’s break down the calculations step by step:
1. Calculate the binomial coefficient [tex]\(\binom{n}{k}\)[/tex]:
[tex]\[ \binom{5}{2} = \frac{5!}{2!(5 - 2)!} = \frac{5!}{2!3!} \][/tex]
[tex]\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \][/tex]
[tex]\[ 2! = 2 \times 1 = 2 \][/tex]
[tex]\[ 3! = 3 \times 2 \times 1 = 6 \][/tex]
So,
[tex]\[ \binom{5}{2} = \frac{120}{2 \times 6} = \frac{120}{12} = 10 \][/tex]
2. Calculate the probability of exactly 2 successes ([tex]\(P(2)\)[/tex]):
[tex]\[ P(2 \text{ successes}) = \binom{5}{2} \cdot p^2 \cdot (1 - p)^{5 - 2} \][/tex]
Substitute [tex]\( \binom{5}{2} = 10 \)[/tex], [tex]\( p = 0.30 \)[/tex], and [tex]\( 1 - p = 0.70 \)[/tex]:
[tex]\[ P(2 \text{ successes}) = 10 \cdot (0.30)^2 \cdot (0.70)^3 \][/tex]
3. Calculate the individual terms:
[tex]\[ (0.30)^2 = 0.09 \][/tex]
[tex]\[ (0.70)^3 = 0.343 \][/tex]
4. Combine them:
[tex]\[ P(2 \text{ successes}) = 10 \cdot 0.09 \cdot 0.343 = 0.3087 \][/tex]
The probability that exactly 2 out of 5 voters will support the ballot initiative is approximately [tex]\(0.3087\)[/tex].
Finally, rounding this to the nearest thousandth gives:
[tex]\[ \boxed{0.309} \][/tex]
Thus, the correct answer from the given choices is:
[tex]\[ \boxed{0.309} \][/tex]
