Question

If X~B(20, 8)
What is the probability that P(X=12)

Answer 1

To find the probability that P(X=12), where X is a random variable following a binomial distribution with parameters n = 20 and p = 8, we can use the formula for the probability mass function of a binomial distribution.

The probability mass function (PMF) of a binomial distribution is given by:

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

Where (n choose k) represents the number of ways to choose k successes out of n trials, p^k represents the probability of k successes, and (1-p)^(n-k) represents the probability of (n-k) failures.

In this case, we want to find the probability P(X = 12), so we substitute n = 20, p = 8, and k = 12 into the formula:

P(X = 12) = (20 choose 12) * (8/20)^12 * (1 - 8/20)^(20-12)

Calculating the values:

(20 choose 12) = 20! / (12! * (20-12)!) = 125970

(8/20)^12 ≈ 0.06871947673

(1 - 8/20)^(20-12) ≈ 0.0011452879

Substituting these values into the formula:

P(X = 12) = 125970 * 0.06871947673 * 0.0011452879

P(X = 12) ≈ 0.0993

Therefore, the probability that P(X = 12) is approximately 0.0993, or 9.93%.