According to an airline, flights on a certain route are on time 85% of the time. Suppose 17 flights are randomly selected and the number of on-time flights is recorded. (a) Explain why this is a binomial experiment. (b) Find and interpret the probability that exactly 12 flights are on time. (c) Find and interpret the probability that fewer than 12 flights are on time. (d) Find and interpret the probability that at least 12 flights are on time. (e) Find and interpret the probability that between 10 and 12 flights, inclusive, are on time.

The formula (nCr)(p^r)(q^(n-r)) will tell us the probability of binomial events occuring. n is the population, r is the desired number of chosen outcomes, p is the probability of success, and q is the probability of failure. nCr tells us how many different ways we can choose r items from a total of n outcomes

Here, n = 17, p = 0.85, q = 0.15 and r depends on the question.

b. r = 12, plug in the values into the formula...

(17C12)(0.85^12)(0.15^5) = 0.0668

c. Use the compliment: the probability of fewer than 12 means 1 - P(12 or more), so 1 - (the sum of the probabilities or 12, 13, 14, 15, 16, or 17 flights being on time). This will save some time when calculating...we have

1 - [ (17C12)(0.85^12)(0.15^5) + (17C13)(0.85^13)(0.15^4) + (17C14)(0.85^14)(0.15^3) + (17C15)(0.85^15)(0.15^2) + (17C16)(0.85^16)(0.15^1) + (17C17)(0.85^17)(0.15^0) ]

= 1 - 0.9681 = 0.0319

d: this is what we just calculated before subtracting from 1 in the last problem, 0.9681

e. This is the probability of 10, 11, or 12 flights being on time

(17C10)(0.85^10)(0.15^7) + (17C11)(0.85^11)(0.15^6) + (17C12)(0.85^12)(0.15^5)

= 0.97