Question

Travelers who fail to cancel their hotel reservations when they have no intention of showing up are commonly referred to as no-shows. In anticipation of no-shows and late cancellations, most hotels overbook(i.e., accept reservations for more rooms than the number of rooms in their inventory). The Journal of Travel Research(Spring 1985) reported that six major hotels in the Seattle area had a no-show rate of10%. Let X equal the number of no-shows among the four travelers who have made hotel reservations in this study.

a) What is the probability that at least two of the four selected will turn to be no-shows?
b) What is the most likely value for X?

Answer 1

a) 0.0523 = 5.23% probability that at least two of the four selected will turn to be no-shows.

b) 0 is the most likely value for X.

Explanation:

For each traveler who made a reservation, there are only two possible outcomes. Either they show up, or they do not. The probability of a traveler showing up is independent of other travelers. This means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

No-show rate of 10%.

This means that
`p = 0.1`

Four travelers who have made hotel reservations in this study.

This means that
`n = 4`

a) What is the probability that at least two of the four selected will turn to be no-shows?

This is
`P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 2) = C_(4,2).(0.1)^(2).(0.9)^(2) = 0.0486`

`P(X = 3) = C_(4,3).(0.1)^(3).(0.9)^(1) = 0.0036`

`P(X = 4) = C_(4,4).(0.1)^(4).(0.9)^(0) = 0.0001`

`P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.0486 + 0.0036 + 0.0001 = 0.0523`

0.0523 = 5.23% probability that at least two of the four selected will turn to be no-shows.

b) What is the most likely value for X?

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(4,0).(0.1)^(0).(0.9)^(4) = 0.6561`

`P(X = 1) = C_(4,1).(0.1)^(1).(0.9)^(3) = 0.2916`

`P(X = 2) = C_(4,2).(0.1)^(2).(0.9)^(2) = 0.0486`

`P(X = 3) = C_(4,3).(0.1)^(3).(0.9)^(1) = 0.0036`

`P(X = 4) = C_(4,4).(0.1)^(4).(0.9)^(0) = 0.0001`

X = 0 has the highest probability, which means that 0 is the most likely value for X.