Because many passengers who make reservations do not show up, airlines often overbook flights (sell more than there are seats). A certain airplane holds 190 passengers. If the airline believes the rate of passenger no-shows is 9% and sells 208 tickets, is it likely they will not have enough seats and someone will get bumped?
[Math] Use the normal model to approximate the binomial to determine the probability of at least 191 passengers showing up
normal distributionprobabilitystatistics
Best Answer
$X$ - numbers of "shows". So, $X \in \mathrm{Bin}(n,p)$ where $n=208$ and $p = 0.91$. Then, if $np(1-p) \geq 10$ we can make the following approximation, where the approximation was made in the last step $$ P_{\mathrm{Bin}}(X \geq 191) = 1- P_{\mathrm{Bin}}(X \leq 190) = 1- P_{\mathrm{Bin}}(X < 191) \approx 1- \mathrm{N}(\frac{190.5-np}{\sqrt{np(1-p)}} ). $$