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 2 people that voted?

Answer 1

To calculate the probability of exactly 2 people voting out of a sample of 7, we can use the binomial probability formula:

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

Where:
- n is the number of trials (sample size)
- k is the number of successes (number of people voting)
- p is the probability of success (proportion of people voting)

In this case, n = 7, k = 2, and p = 0.39.

First, let's calculate the binomial probability:
P(X = 2) = C(7, 2) * (0.39)^2 * (1-0.39)^(7-2)

The combination function C(7, 2) can be calculated as:
C(7, 2) = 7! / (2! * (7-2)!)

= 7! / (2! * 5!)
= (7 * 6 * 5!)/ (2! * 5!)
= (7 * 6) / (2 * 1)
= 21

Now, we can substitute these values into the probability formula:
P(X = 2) = 21 * (0.39)^2 * (1-0.39)^(7-2)
P(X = 2) = 21 * 0.39^2 * 0.61^5
P(X = 2) ≈ 0.3196

Therefore, the probability of exactly 2 people voting out of a sample of 7 is approximately 0.3196, or 31.96%.