According to a survey, every 3 out of 10 young adults in the US has been to a mu

Question

According to a survey, every 3 out of 10 young adults in the US has been to a music festival before. In a randomly selected group of 8 young adults:

a) What is the probability that exactly 3 of them have been to a music festival?

b) What is the probability that exactly 3 of them haven’t been to a music festival? (This is NOT the complement of (a)!)

c) What is the probability that at least 5 of them have been to a music festival?

d) What is the probability that at most 3 of them have been to a music festival?

e) What is the probability that between 1 and 3 (inclusive) of them have been to a music festival?

Explanation / Answer

p = 3/10 = 0.3

n = 8

P(X = x) = 8Cx * 0.3x * (1 - 0.3)8-x

a) P(X = 3) = 8C3 * 0.33 * 0.75 = 0.2541

b) p = 1 - 0.3 = 0.7

P(X = 3) = 8C3 * 0.73 * 0.35 = 0.0467

c) P(X > 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)

                  = 8C5 * 0.35 * 0.73 + 8C6 * 0.36 * 0.72 + 8C7 * 0.37 * 0.71 + 8C8 * 0.38 * 0.70

                  = 0.8059

d) P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

                  = 8C0 * 0.30 * 0.78 + 8C1 * 0.31 * 0.77 + 8C2 * 0.32 * 0.76 + 8C3 * 0.33 * 0.75

                  = 0.058

e) P(1 < X < 3) = P(X = 1) + P(X = 2) + P(X = 3)

                         = 8C1 * 0.31 * 0.77 + 8C2 * 0.32 * 0.76 + 8C3 * 0.33 * 0.75

                         = 0.058

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