A In the fall it has been observed that three consecutive we

(A) In the fall it has been observed that three consecutive weeks of september had no thefts. Is this suprising, justify your answer analytically.

(B) Would if be suprising if one found that in each of those three weeks the number of thefts were greater than 5?

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 =    3.5      
          
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.030197383

Thus, for 3 weeks of no theft,

P(3 weeks no theft) = 0.030197383^3 = 2.75364*10^-5   

As this is a very low probability, yes, this is suprising. [ANSWER]

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b)

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 =    3.5      
          
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   ) =    0.857613553
          
Thus, the probability of at least   6   successes is  
          
P(more than   5   ) =    0.142386447

Thus, for 3 consecutive weeks,

P(3 weeks, more than 5) = 0.142386447^3 = 0.002886729

This is surprising, as this is also a very low probability.