Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.
Sagot :
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]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
