Recently the movement of orders has not gone as planned and

Recently, the movement of orders has not gone as planned, and there were a large number of complaints. Bud Owens, director of customer service, has completely redone the method of order handling. The goal is to have fewer than 5 unfilled orders on hand at the end of 90% of the working days. Frequent checks of the unfilled orders at the end of the day reveal that the distribution of unfilled orders follows a Poisson distribution with a mean of two orders. Has New Process, Inc. lived up to its internal goal? Mr. Owens wants some evidence to support your answer.

Please work the calculations out so I actually know how to do the problem!

Solution

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 =    2      
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   ) =    0.947346983
          
Which is also          
          
P(fewer than   5   ) =    0.947346983 [PROBABILITY]

As this is greater than 90%, then YES, it has lived up to its internal goal. [ANSWER]