Suppose for a biased dice the probability of showing a six i

Suppose for a biased dice, the probability of showing a six in one toss is 4/15. If the die isthrown 12 times, finda. The probability that a six will occur exactly twice.
b. The probability that six occur less than four times.c. The expected number of sixes.

Solution

A)

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 =    0.266666667      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.211103611 (answer)

B)

Note that P(fewer than x) = P(at most x - 1).          
          
Using a cumulative binomial distribution table or technology, matching          
          
n = number of trials =    12      
p = the probability of a success =    0.266666667      
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   ) =    0.596727535
          
Which is also          
          
P(fewer than   4   ) =    0.596727535 (answer)

C)

Mean = n p =12(4/15) =3.2 (answer)