A biased die is thrown 30 times and the number of sixes seen

A biased die is thrown 30 times and the number of sixes seen is eight. If the die is thrown a further twelve times find: the probability that a Six will occur exactly twice?

Solution

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    12      
p = the probability of a success = 8/30 =    0.266666667      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.211103611