# Expected number of passengers for the flight

One of the interview questions that I was asked was :

The small commuter plane has 30 seats. The probability that any particular passenger will not show up for a flight is 0.1, independent of other passengers.

Design a function to get largest m value such that expectedshow[m] = expected number of passengers showing up if 30 + m tickets are sold for a flight and expectedshow[m] ≤ 30.

I could not think of a way to approach this question. Can somebody explain it ?

• First $(30+m)\cdot0.9\leq30\implies0.9m\leq3\implies{m}\leq3.3$. Then $[m\in\mathbb{N}]\wedge[m\leq3.3]\implies\max(m)=3$. – barak manos Nov 12 '14 at 16:26

If $n$ tickets are sold, then the expected number of passengers that will show is $0.9n$, by linearity of expectation.
However, in this problem the variable is not $n$, but $m$, where $n=30+m$. Hence the number of expected passengers that will show is $0.9(30+m)$. Now consider those values of $m$ that make this at most 30; choose the largest. You should get $$\frac{30}{0.9}-30=3.\overline{3}$$ Most likely the interviewer is looking for the floor of this, namely $m=3$.