Solved – Calculating Poisson probability for persons’ arrival at a doctor’s clinic

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Patients arrive randomly and independently at a doctor's clinic from 8·0 A.M. at an average rate of one in Five minutes.The waiting room holds 12 persons. What is the probability that the room will be full when the doctor arrives at 9·0 A.M.?

I know how to solve some Poisson distribution based problem. But for this one I am not sure what am I missing. Can someone please help to exactly where am I wrong.
According to me:
Lambda = 12/hour.
Number of events (desired)= 12
so Probability(12 arrivals in an hour) = e^(-12)*(12^12)/(12!). (not confident in this method though) Please help.

Best Answer

@whuber is right, the probability of the event "the room is full" is the probability of 12 persons or more entering the waiting room. This is the same as calculating the complement of the event of 11 persons. Below I illustrate with some R code. 

# rate is 1/5 = 12/60
d <- 12

# set lower.tail = FALSE to calculate the complement event
p <- ppois(q = 11, lambda = d, lower.tail = FALSE)
format(p, scientific = FALSE)
# [1] "0.5384027"
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