In a small town out of 12 accidents that occurred in June 19

In a small town, out of 12 accidents that occurred in June 1986, four happened on Friday the 13th. Is this a good reason for a superstitious person to argue that Friday the 13th is unlucky? [Hint: Assume that the probability of an accident on Friday the 13th is 1/30; note that this is not a question about whether exactly 4 occurred, but something slightly more complicated.]

Solution

Here, we get the probability of at least 4 accidents. How rare is it?

Note that P(at least x) = 1 - P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    30      
p = the probability of a success =    0.033333333      
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   ) =    0.98306566
          
Thus, the probability of at least   4   successes is  
          
P(at least   4   ) =    0.01693434   

0.01693434 is a rare ocurrence (depending on what standards you set as \"rare\"), so it could be good reason that Friday the 13th is unlucky this way.