In 2002, approximately 39% of the people eligible to vote in mid-term elections voted. Suppose you have taken a sample of 7 eligible voters. What is the probability of having sampled exactly 3 people that voted?

Answer 1

To calculate the probability of exactly 3 out of 7 eligible voters voting, we use the binomial probability formula:

P(X=k) = (n choose k) * p^k * (1-p)^(n-k)

Where:
- n is the number of trials or eligible voters sampled (7)
- k is the number of successful outcomes or voters that voted (3)
- (n choose k) is the number of ways to choose k successes out of n trials (calculated as n! / (k! * (n-k)!))
- p is the probability of a successful outcome or the proportion of eligible voters that vote (0.39)
- (1-p) is the probability of a failure (1-0.39)

Plugging in the values:

P(X=3) = (7 choose 3) * 0.39^3 * (1-0.39)^(7-3)

Calculating the binomial coefficient:

(7 choose 3) = 7! / (3! * (7-3)!)
= 7! / (3! * 4!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35

Calculating the probability:

P(X=3) = 35 * 0.39^3 * (1-0.39)^(7-3)
= 35 * 0.39^3 * 0.61^4
≈ 0.2812

Therefore, the probability of having sampled exactly 3 people that voted is approximately 0.2812, or 28.12%.