To determine the probability that exactly 2 out of 5 randomly surveyed voters will support the ballot initiative from a population where 3046 out of 3046 voters support it, we use the binomial probability formula:
[tex]\[ P(k \text{ successes }) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
Here,
- [tex]\( n \)[/tex] is the sample size, which is 5 in this case.
- [tex]\( k \)[/tex] is the number of favorable outcomes we are looking for, which is 2.
- [tex]\( p \)[/tex] is the probability of a single voter supporting the initiative.
First, calculate [tex]\( p \)[/tex]:
[tex]\[ p = \frac{\text{total voters support}}{\text{total population}} = \frac{3046}{3046} = 1 \][/tex]
Given [tex]\( p = 1 \)[/tex], the probability of a single voter not supporting the initiative (1 - p) would be:
[tex]\[ 1 - p = 1 - 1 = 0 \][/tex]
Now, we need to calculate [tex]\( \binom{n}{k} \)[/tex], also known as "n choose k":
[tex]\[ \binom{n}{k} = \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} = \frac{5 \cdot 4 \cdot 3!}{2 \cdot 1 \cdot 3!} = \frac{5 \cdot 4}{2 \cdot 1} = 10 \][/tex]
We now use the binomial probability formula:
[tex]\[ P(2 \text{ successes }) = \binom{5}{2} (1)^2 (0)^{5-2} \][/tex]
Since anything raised to the power of zero is 1, this simplifies to:
[tex]\[ P(2 \text{ successes }) = 10 \cdot 1^2 \cdot 0^3 = 10 \cdot 1 \cdot 0 = 0 \][/tex]
So, the probability that exactly 2 voters out of 5 will be in favor of the ballot initiative is:
[tex]\[ P = 0 \][/tex]
When rounded to the nearest thousandth, this probability remains:
[tex]\[ \boxed{0.0} \][/tex]
This result means that it is essentially impossible to find exactly 2 supporters in a sample of 5 if every single voter in the population supports the initiative (as confirmed by our specific dataset).
