for the following example of an "Overbooking" I have to calculate the probability by using the Central Limit Theorem:
- An airline books 52 seats whilst there are only 50 seats available. A guest appears with a probability of 90%. What's the probablity that maximum 50 guests are coming?
I thought about normal approximation:
- $P(0 \leq S_n \leq 50) \approx \phi (\frac{50+0.5-µ}{\sigma}) - \phi (\frac{0-0.5-µ}{\sigma})$.
- $µ = np = 52\cdot 0.1 = 5.2$.
- $\sigma = \sqrt{npq} = \sqrt{4.68}$
Finally I get $0.99978$, whereas the probability by using binomial distribution gives me $0.9717$.
Have I done it correctly or are there some mistakes?