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 exactly 3 people that voted?

Answer 1

To find the probability of exactly 3 people out of 7 having voted, we need to use the binomial probability formula. The formula is:

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

Where:
P(X = k) is the probability of getting exactly k successes (in this case, 3 people voted).
n is the total number of trials (sample size, which is 7 voters in this case).
k is the number of successful outcomes (people who voted, which is 3 in this case).
p is the probability of success in a single trial (39% or 0.39 since 39% of people voted).
(1 - p) is the probability of failure in a single trial (1 - 0.39 = 0.61).

Using this formula, we can calculate the probability:

P(X = 3) = (7 choose 3) * 0.39^3 * 0.61^(7 - 3)

First, let's calculate the combination (n choose k):

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

Next, we plug these values into the formula:

P(X = 3) = 35 * 0.39^3 * 0.61^(7 - 3)
= 35 * 0.39^3 * 0.61^4 ≈ 0.2329

Therefore, the probability of exactly 3 out of 7 people having voted is approximately 0.2329, or 23.29%.