A man sees on average 38 customers per day What is the proba

A man sees on average 3.8 customers per day. What is the probability that in one day he sees:

a) 5 customers?

b) less than 4 customers?

c) more than 5 customers?

d) at most 6?

Solution

a)

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 =    3.8      
          
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.147712656 [ANSWER]

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b)

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 =    3.8      
          
x = our critical value of successes =    4      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   3   ) =    0.473484843
          
Which is also          
          
P(fewer than   4   ) =    0.473484843 [ANSWER]

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c)

Note that P(more than x) = 1 - P(at most x).          
          
Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    3.8      
          
x = our critical value of successes =    5      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   5   ) =    0.815556256
          
Thus, the probability of at least   6   successes is  
          
P(more than   5   ) =    0.184443744 [ANSWER]

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d)

Using a cumulative poisson distribution table or technology, matching          
          
u = the mean number of successes =    3.8      
          
x = the maximum number of successes =    6      
          
Then the cumulative probability is          
          
P(at most   6   ) =    0.909107605 [ANSWER]