You want to know the probability that exactly 8 out of 9 institutions offer distance learning courses, given that the success probability of an institution offering a distance learning course is 94%, or 0.94.
The binomial distribution model is a suitable tool for this problem. It describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success.
The formula for the binomial distribution probability mass function (PMF) is:
P(x;n,p) = C(n, x) * (p^x) * ((1-p)^(n-x))
Where:
- P(x;n,p) is the probability of getting exactly x successes in n trials,
- C(n, x) is the binomial coefficient ("n choose x"),
- p is the probability of success in a single trial (in this case, 0.94),
- (p^x) is the probability of getting exactly x successes,
- ((1-p)^(n-x)) is the probability of getting exactly (n-x) failures.
Plugging in the numbers in the above formula, we have:
P(8;9,0.94) = C(9, 8) * (0.94^8) * ((1-0.94)^(9-8))
Calculate the binomial coefficient C(9, 8) and substitute the values into the formula to obtain the probability of exactly 8 institutions offering distance learning courses.
The result is approximately 0.3292. So, the probability that exactly 8 out of 9 institutions offer distance learning courses is 0.3292, or 32.92%.