Question

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.

Answer 1

Answer:

a:  It is binomial because it is either on time, or it's not. There are only 2 choices

b:  0.0668

c:  0.0319

d:  0.9681

e:  0.097

Step-by-step explanation:

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