A recent poll found that 45% of eligible voters are planning to vote in favor of a new by-law. Suppose you randomly survey six voters. What is the probability that at least three of the voters plan to vote in favor of the new by-law? (1 point) Responses 13.2% 13.2% 25.5% 25.5% 30.3% 30.3% 55.8% 55.8%
In order to calculate the probability that at least three of the six voters plan to vote in favor of the new by-law, we need to consider the different ways in which this can occur.
The probability that exactly three voters plan to vote in favor of the new by-law is given by the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- n is the total number of trials (6 voters)
- k is the number of successful trials (3 voters in favor)
- p is the probability of success on a single trial (45% or 0.45)
Calculating for exactly 3 voters in favor:
P(X = 3) = (6 choose 3) * 0.45^3 * (1-0.45)^(6-3)
= 20 * 0.45^3 * 0.55^3
= 0.3364 or 33.64%
The probability of exactly 4 voters in favor would be:
P(X = 4) = (6 choose 4) * 0.45^4 * 0.55^2
= 15 * 0.45^4 * 0.55^2
= 0.2273 or 22.73%
The probability of exactly 5 voters in favor would be:
P(X = 5) = (6 choose 5) * 0.45^5 * 0.55^1
= 6 * 0.45^5 * 0.55
= 0.0769 or 7.69%
The probability of all 6 voters in favor would be:
P(X = 6) = (6 choose 6) * 0.45^6 * 0.55^0
= 1 * 0.45^6 * 1
= 0.0119 or 1.19%
Therefore, the probability that at least three of the six voters plan to vote in favor of the new by-law is the sum of the probabilities of exactly 3, 4, 5, and 6 voters in favor:
P(X >= 3) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)
= 33.64% + 22.73% + 7.69% + 1.19%
= 65.25% or 65.25%
Therefore, the correct answer is 65.25%.