There are on average 8 visits per hour to an antique store 1

There are on average 8 visits per hour to an antique store.

1) What is the probability of fewer than 2 visits will be made in the next half hour?

2) What is the probability of more than 2 visits will be made in the next half hour?

3) What is the probability of at least 1 visit will be made in the next 15 minutes?

Solution

1.

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 = 8(1/2 hr) =   4      
          
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x - 1) from a table/technology is          
          
P(at most   1   ) =    0.091578194
          
Which is also          
          
P(fewer than   2   ) =    0.091578194 [ANSWER]

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2.

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 = 8(1/2 hr)   4      
          
x = our critical value of successes =    2      
          
Then the cumulative probability of P(at most x) from a table/technology is          
          
P(at most   2   ) =    0.238103306
          
Thus, the probability of at least   3   successes is  
          
P(more than   2   ) =    0.761896694 [ANSWER]

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3.

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 = 8(1/4 hr) =   2      
          
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.135335283

P(at least 1) = 1 - P(0) = 0.864664717 [ANSWER]