In a hotel, the average number of customers arriving during the morning hours on a weekday is 10. Find the probability that

Question

In a hotel, the average number of customers arriving during the morning hours on a weekday is 10. Find the probability that

(a) there are 4 or more customers arriving during the morning hours on a certain weekday.

(b) there are exactly 4 customers arriving during the morning hours on a certain weekday.

(c) during a normal 5-day working week, there will be exactly 2 days on which there are exactly 4 customers arriving during the morning hours.

Answer 1

So from the looks of this, one may see that it is indeed a Poisson Distribution where λ=10.

Thus, for any x, we know P(x) = (e^-10*10^x)/x!.

(a) is asking P(X≥4).

P(X≥4) = 1–P(X<4) = 1–[P(3) + P(2) + P(1) + P(0)]

= 1–[(e^-10*10³)/3! + (e^-10*10²)/2! + (e^-10*10¹)/1! + (e^-10*10⁰)/0!]

= 1–[0.0076+0.0023+0.0005+0]

= 0.9896

(b) is asking P(X=4).

P(4) = (e^-10*10⁴)/4! = .0189

(c) is asking what is the probability 2 out of the 5 days, there will be four customers arriving.

To figure this out, we may use the binomial formula.

n=5 (for 5 days)

x=2 (2 successes in the 5 days)

p=.0189 (from part (b))

(n choose x)*p^x*(1-p)^n-x

(5 choose 2)*.0189²*(1-.0189)^5-2

.0034