Question

A dance studio teaches students a contemporary dance form. It is observed that the probability of students from this studio becoming dance teachers is 0.69. An event management company wants to use this studio for a show. They decide to observe it for a year before making the deal. They enroll 120 students of their own in this studio and decide that once 75 of them get jobs as dance teachers, they will sign the contract with them. What is the probability that the event management company will sign the contract with the dance studio?

Answer 1

A. 0.076
B. 0.053
C. 0.024
D. 0.924
E. 0.947

Answer 2

Challenging problem. First, note that the expect number of successes is Np = 120(0.69) = 82.8 become teachers. This helps rule out A,B,C. the standard deviation for the binomial distribution, which this is, is sqrt[N(p)(1-p)] = sqrt(25.7) = 5.1

The goal of 75 is exceeded by the expected value 82.8, but only a difference of 7.8, which is less than two standard deviations, so E is a bit high and D seems likely to be the answer.