probability
did you know airlines overbook flights on purpose?
a predictable share of passengers never show up, so airlines sell the empty seats twice — and probability tells them how far to push it.
expected to show
179
of 180 seats
expected empty seats
1.4
revenue left behind
chance someone is bumped
27.3%
more show than seats
push it this far and gate agents start sweating — compensation costs eat the gains.
the idea
on almost every flight, some passengers don’t show. an empty seat is money gone forever, so airlines sell a few extra tickets and let probability manage the risk.
how it works
each passenger is a coin flip with a known-ish probability of not showing. chains of flips follow the binomial distribution, which tells the airline the chance that more people show up than there are seats.
go deeper
selling the same seat twice, carefully
if a 180-seat plane historically sees about 5% of passengers not turn up, selling exactly 180 tickets means flying with empty seats most days.
selling, say, 188 tickets means the plane usually still fits everyone — and the airline keeps the extra revenue.
expected value runs the numbers
each passenger is a coin flip with a known-ish probability of not showing. chains of flips follow the binomial distribution, which tells the airline the chance that more people show up than there are seats.
the airline weighs the certain extra revenue from each extra ticket against the cost of compensating the rare bumped passenger — and picks the number of extra tickets that maximises expected profit.
expected no-shows = tickets sold × P(no-show)
a binomial bet
if each of n passengers shows up independently with probability p, the number who show follows a binomial distribution with mean n·p.
sell 188 tickets at a 95% show rate and you expect about 179 passengers — just under the 180 seats. but the distribution has a spread, so there’s still a small chance of 181 or more turning up.
overbooking models choose n so that P(show-ups > seats) stays acceptably small while expected revenue climbs.
P(k show up) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
things to wonder about next
- when everyone does show up, airlines ask for volunteers first — compensation is part of the expected cost, not an accident.
- show-up rates vary by route, day and even weather, so the models are re-fitted constantly.
- the same expected-value thinking sets hotel overbooking, clinic appointment slots and restaurant table spacing.
sources & further reading
concepts: expected value · binomial probability · optimisation
5 minute read