binomial &poisson distribution

Question

on an average  A can solve 40% of the problems.what is the probability of A solving[a]. no problems out of 6. [b]exactly four problems out of 6

Answer 1

Binomially, if p=0.4 and (1-p)=0.6.

(p+(1-p))^6=1 is the expression for the combination of probabilities.

(a) For solving no problems out of 6, we have 0.6^6=0.046656 approx, or 4.6656%.

(b) For exactly 4 out of 6 we need to find the coefficients of the expansion first: 1,6,15,20,15,6,1. We need 15 to apply to  p^4(1-p)^2, so we have 15(0.4)^4(0.6)^2 (4 successes and 2 failures)=0.13824 or 13.824%.

Poisson distributions are normally applied to events occurring in time. This problem is a simple binomial (2-state problem: success or failure) so the binomial distribution is applicable.