For a binomial random variable X with n 25 and p 4 evalua

. For a binomial random variable, X, with n = 25 and p = .4, evaluate in the easiest manner possible

P(6 X 12).

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X < = 12) = P(X=12) + P(X=11) + P(X=10) + P(X=9) + P(X=8) + P(X=7) + .....P(X=1) + P(X=0)
= ( 25 12 ) * 0.4^12 * ( 1- 0.4 ) ^13 + ( 25 11 ) * 0.4^11 * ( 1- 0.4 ) ^14 + ( 25 10 ) * 0.4^10 * ( 1- 0.4 ) ^15 + ( 25 9 ) * 0.4^9 * ( 1- 0.4 ) ^16 + ( 25 8 ) * 0.4^8 * ( 1- 0.4 ) ^17 + ( 25 7 ) * 0.4^7 * ( 1- 0.4 ) ^18 + .....
+ ( 25 1 ) * 0.4^1 * ( 1- 0.4 ) ^24 + ( 25 0 ) * 0.4^0 * ( 1- 0.4 ) ^25
= 0.8462

P( X < 6) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0) +
= ( 25 5 ) * 0.4^5 * ( 1- 0.4 ) ^20 + ( 25 4 ) * 0.4^4 * ( 1- 0.4 ) ^21 + ( 25 3 ) * 0.4^3 * ( 1- 0.4 ) ^22 + ( 25 2 ) * 0.4^2 * ( 1- 0.4 ) ^23 + ( 25 1 ) * 0.4^1 * ( 1- 0.4 ) ^24 + ( 25 0 ) * 0.4^0 * ( 1- 0.4 ) ^25 +   
= 0.0294

P( 6 <= X <= 12) = 0.8462 - 0.0294 = 0.8168