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94% of the largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 9 such intitutions, and let X be the number of intitutions that offer distance learning courses. Please show the following answers to 4 decimal places. a. What is the probability that exactly 8 intitutions in this study offer distance learning courses?

User Blow
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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%.

User Jtm
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