A man has a hotel with 30 rooms but he offers 32 rooms because 10% of people who booked a room won't show up.
a) Probability, that there will be overbookings: P(31 people show up) + P(32 people show up) = $ \binom{32}{31} * 0.9^{31} * 0.1^1 + \binom{32}{32} * 0.9^{32}*0.1^0 = 0.16$. This is pretty easy.
b) Calculate the expected profit if $X$ means the number of overbooked rooms and the profit is described via the function $Prof(X) = 4800 - 500X$. I don't know how to solve this. I tried to somehow distinguish between $X=0$ and $X\gt0$, but it doesn't work out. The solution should be 4705.
c) Should he offer 30, 31, 32 or 33 beds to maximize the expected profit?
Thanks.
I hope that I will help you.
b) Probability that there will be exactly one overbooking is ${{32}\choose{31}} \cdot 0.9^{31} \cdot 0.1 = 0.122$ and probability that there will be 2 overbookings is $0.9^{32} = 0.034$ (as you correctly stated in a)). This means that probability that $X = 1$ is 0.122, probability that $X = 2$ is 0.034 and probability for $X = 0$ (no overbooking) is $1 - 0.122 - 0.034 = 0.844$. So the expected profit is $EX = 0.844*4800 + 0.122*4300 + 0.034*3800 = 4705$.
I gave you the hint how to calculate expected profit also for 30, 31 and 33 beds which is required in c). Are you able to do so or should I help you more?