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A restaurant sees 70 customers in a 24 period based on historical patterns. However, 40 customers came in today between 9AM-11AM. What is the probability that in the next 22 hours that there will be atleast 30 more customers?

The assumption that the customer walk ins are uniformly distributed. Meaning for e.g. if the limit was 48 customers, we observe that there is two customers per hour uniformly across the day & night

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This is one of those all too frequent problems where one is expected to make an assumtion that is patently unreasonable. Perhaps someone wrote a glowing review on Placebook. – André Nicolas Oct 1 '12 at 20:23

It doesn't matter what happened from 9:00 to 11:00. The probability of seeing at least $30$ customers in the next $22$ hours is the same as usual. If you assume that the arrivals are Poisson-distributed, where the expected number in $22$ hours is $\lambda = 70\cdot 22/24 = 385/6 \approx 64.17$, then the probability of seeing at least $30$ customers in $22$ hours is $$ P[X\ge 30] = \sum_{k=30}^{\infty} \frac{\lambda^k e^{-\lambda}}{k!} = 1 - \frac{\Gamma(30, \lambda)}{\Gamma(30)}\approx1-7.0825\times 10^{-7}. $$ In other words, the busy morning almost guarantees a good day overall.

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Thanks @ mjqxxxx - the answer is correct and based on Poisson distribution and can also be validated here - – Joe Oct 2 '12 at 14:27

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