What is the probability that 3 of 8 randomly selected individuals will have exceeded the insurance deductible, i.e. that 5 of 8 will not exceed the deductible? Recall that 70% of individuals will not exceed the deductible.
We would like to apply the binomial model, so we check the conditions. The number of trials is fixed (n = 8) (condition 2) and each trial outcome can be classified as a success or failure (condition 3). Because the sample is random, the trials are independent (condition 1) and the probability of a success is the same for each trial (condition 4).
In the outcome of interest, there are k = 5 successes in n = 8 trials (recall that a success is an individual who does not exceed the deductible, and the probability of a success is p = 0.7. So the probability that 5 of 8 will not exceed the deductible and 3 will exceed the deductible is given by(
8
5
) (0.7)5(1− 0.7)8−5 =
8!
5!(8− 5)! (0.7)5(1− 0.7)8−5
= 8!
5!3! (0.7)5(0.3)3