Lesson Binomial distribution problems on overbooking flights

Binomial distribution problems on overbooking flights

Problem 1

It is a binomial experiment.


There are 11 potential passengers. The probability that every given passenger will show up is 0.9.

The events for individual passengers are INDEPENDENT.

The question is: find the probability that of 11 passengers that bought tickets, at least 10 will show up
(it is the event "the flight is overbooked").


n = 11 (the number of trials);

p = 0.9  (the probability of "success" to each individual trial);

k >= 10  (the event of overbooking flight).


P(n=11; p=0.9; k>=10) = P(n=11; p=0.9; k=10) + P(n=11; p=0.9; k=11) =  +  =

   =  = 0.697  (rounded).    ANSWER

Problem 2

                 Question  (a)


It is a binomial experiment.


There are 59 potential passengers. The probability that every given passenger will show up is 1 - 0.0999 = 0.9001.

The events for individual passengers are INDEPENDENT.

The question is: find the probability that of 59 passengers that bought tickets, 58 or 59 will show up
(it is the event "the flight is overbooked").


n = 59 (the number of trials);

p = 0.9001  (the probability of "success" to each individual trial);

k >= 58  (the event of overbooking flight).


P(n=59; p=0.9001; k>=58) = P(n=59; p=0.9001; k=58) + P(n=59; p=0.9001; k=59) =  +  =

   =  = 0.01316  (rounded).    ANSWER



                 Question  (b)


It is a binomial experiment very similar to part (a).


There are 63 potential passengers. The probability that every given passenger will show up is 1 - 0.0999 = 0.9001.

The events for individual passengers are INDEPENDENT.

The question is: find the probability that of 63 passengers that bought tickets, at least 58 will show up
(it is the event "the flight is overbooked").


n = 63 (the number of trials);

p = 0.9001  (the probability of "success" to each individual trial);

k >= 58  (the event of overbooking flight).


We want to find the probability  P = P(n=63; p=0.9001; k>=58) 


Go to web-site https://stattrek.com/online-calculator/binomial and use online (free of charge) calculator there.
Its interface is very simple, so even a beginner student can easily work with it.
It facilitate using  many-addend formula, so calculations are really easy and fast.


The ANSWER is

       P = P(n=63; p=0.9001; k>=58)  = 0.3893.   


Alternatively, you may use standard function binomcfd on your regular TI-83/84 calculator.

About this function and how to use it read from this web-page https://www.statology.org/binompdf-vs-binomcdf/

The regular calculator will provide the same answer.