Let's solve the problem step-by-step to determine the probability that exactly 2 voters out of 5 will be in favor of the ballot initiative, given that 30% of voters support it.
Given values:
- Total number of voters surveyed, [tex]\( n = 5 \)[/tex]
- Probability of success (a voter supporting the initiative), [tex]\( p = 0.30 \)[/tex]
- Number of successes (voters in favor), [tex]\( k = 2 \)[/tex]
We need to calculate the binomial probability, which is given by the formula:
[tex]\[ P(k \text{ successes}) = {}_nC_k \cdot p^k \cdot (1 - p)^{n - k} \][/tex]
Where:
[tex]\[ {}_nC_k = \frac{n!}{(n - k)! \cdot k!} \][/tex]
is the binomial coefficient.
Let's break it down into steps:
### Step 1: Compute the binomial coefficient [tex]\({}_nC_k\)[/tex]
Using:
[tex]\[ {}_nC_k = \frac{n!}{(n - k)! \cdot k!} \][/tex]
we substitute [tex]\( n = 5 \)[/tex] and [tex]\( k = 2 \)[/tex]:
[tex]\[ {}_5C_2 = \frac{5!}{(5 - 2)! \cdot 2!} = \frac{5!}{3! \cdot 2!} \][/tex]
Calculating factorials:
[tex]\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \][/tex]
[tex]\[ 3! = 3 \times 2 \times 1 = 6 \][/tex]
[tex]\[ 2! = 2 \times 1 = 2 \][/tex]
Now, compute the binomial coefficient:
[tex]\[ {}_5C_2 = \frac{120}{6 \times 2} = \frac{120}{12} = 10 \][/tex]
### Step 2: Compute [tex]\( p^k \)[/tex] and [tex]\( (1 - p)^{n - k} \)[/tex]
Given [tex]\( p = 0.30 \)[/tex]:
[tex]\[ p^k = (0.30)^2 = 0.30 \times 0.30 = 0.09 \][/tex]
Given [tex]\( 1 - p = 0.70 \)[/tex]:
[tex]\[ (1 - p)^{n - k} = (0.70)^{5 - 2} = (0.70)^3 = 0.70 \times 0.70 \times 0.70 = 0.343 \][/tex]
### Step 3: Combine all parts to find the probability
Now, use the binomial probability formula:
[tex]\[ P(k \text{ successes}) = {}_nC_k \cdot p^k \cdot (1 - p)^{n - k} \][/tex]
Substitute the computed values:
[tex]\[ P(2 \text{ successes}) = 10 \cdot 0.09 \cdot 0.343 \][/tex]
Perform the multiplication:
[tex]\[ P(2 \text{ successes}) = 10 \cdot 0.03087 = 0.3087 \][/tex]
### Step 4: Round to the nearest thousandth
To round [tex]\( 0.3087 \)[/tex] to the nearest thousandth:
[tex]\[ 0.3087 \approx 0.309 \][/tex]
So, the probability that exactly 2 voters out of 5 will be in favor of the ballot initiative is [tex]\( \boxed{0.309} \)[/tex].
