Question

A popular resort hotel has 300 rooms and is usually fully booked. About 4% of the time a reservation is cancelled before 6:00 PM deadline with no penalty. What is the probability that at least 280 rooms will be occupied? Use binomial distribution to find the exact value and the normal approximation to the binomial and compare your answers.

Answer 1

Using Binomial probability function, we are looking for:
P(280≤X≤300)
=Σ C(300,r)p^r*(1-p)^(300-r)
for r=280, 281, ...300, and
C(300,r)=300!/((300-r)!r!)
=0.00896+0.01530+0.02474+0.03777+0.05427+...+0.00004+0.0000
=0.98992

Using normal approximation with continuity correction, we have
n=300, p=0.96,μ=np=288,σ=sqrt(npq)=3.3941
P(280≤X≤300)
=1-Φ(279.5,288,3.3941)
=0.99386
Error = (0.99386-0.98992)/0.98992=0.4%.

If we look at the tail, the performance is much worse. The reason for the poor performance (if we look at the tail) is because
np=300*0.96=288 >> 10
npq=300*0.96*0.04= 11.52 ~ 10.

If we use Camp Paulson approximation, we get
1-P(279≤X≤300)
=0.98982 (error=-0.0096%, much better)

Ref:
http://www.johndcook.com/blog/camp_paulson/