Statistics Question
If X ~ Bin (n, p), then the probability mass function of X is:
p (x) = (nCx) px (1 – p)n – x; for x = 0, 1, …, n; 0 < p <1; p (x) is 0 otherwise.
Calculation:
Consider the event that a passenger boards a flight as a success. Then, missing the flight can be considered as failure.
Here, probability of missing a flight is 0.0968. This is the probability of failure.
Thus, the probability of success is, p = 0.9032 (= 1 – 0.0968).
(a)
Denote X as the number of successes in n = 55 trials.
The probability that 54 or 55 passengers show up for the flight is:
P (X = 54) + P (X = 55)
= p (54) + p (55)
= 0.0218 + 0.0037
[Using the formula in the previous step, or by Excel formulae:
=BINOM.DIST(54,55,0.9032,FALSE) for p (54),
=BINOM.DIST(54,55,0.9032,FALSE) for p (55)].
= 0.0255.
Thus, if 55 tickets are sold, the probability that 54 or 55 passengers show up for the flight resulting in an overbooked flight is 0.0255.
(b)
Denote X as the number of successes in n = 59 trials.
A passenger will have to be bumped, if more than 53 passengers show up for the flight. The relevant probability is calculated below:
P (X > 53)
= 1 – P (X ≤ 53)
= 1 – 0.513
[Using Excel formula =BINOM.DIST(53,59,0.9032,TRUE)]
= 0.487.
Thus, if 59 tickets are sold, the probability that a passenger will have to be "bumped" is 0.487.
[We have interpreted the meaning of “a passenger” to be any number of passengers more than 53].
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