Customers arrive at a bakery at an average rate of 10 custom

Customers arrive at a bakery at an average rate of 10 customers per hour. What is the probability that less than 5 customers will arrive in the next hour? Assume Poisson arrivals.

Customers arrive at a bakery at an average rate of 10 customers per hour. What is the probability that exactly 20 customers will arrive in the next 2 hours? Arrivals are Poisson.

Solution

a)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    10      
p = the probability of a success =    0      
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   4   ) =    0.029252688
          
Which is also          
          
P(fewer than   5   ) =    0.029252688 [ANSWER]

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b)

The mean number of customers in 2 hrs is 10*2 = 20 customers.

Note that the probability of x successes out of n trials is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes =    20      
          
x = the number of successes =    20      
          
Thus, the probability is          
          
P (    20   ) =    0.088835317 [ANSWER]