If n 5 and 40 what is the probability that X 4 X 3 X 2 X

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

a)
P( X = 4 ) = ( 5 4 ) * ( 0.4^4) * ( 1 - 0.4 )^1
= 0.0768
b)
P( X < = 3) = P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 5 3 ) * 0.4^3 * ( 1- 0.4 ) ^2 + ( 5 2 ) * 0.4^2 * ( 1- 0.4 ) ^3 + ( 5 1 ) * 0.4^1 * ( 1- 0.4 ) ^4 + ( 5 0 ) * 0.4^0 * ( 1- 0.4 ) ^5
= 0.913

c)
P( X < 2) = P(X=1) + P(X=0)
= ( 5 1 ) * 0.4^1 * ( 1- 0.4 ) ^4 + ( 5 0 ) * 0.4^0 * ( 1- 0.4 ) ^5   
= 0.337
d)
P( X < = 1) = P(X=1) + P(X=0)
= ( 5 1 ) * 0.4^1 * ( 1- 0.4 ) ^4 + ( 5 0 ) * 0.4^0 * ( 1- 0.4 ) ^5
= 0.337
P( X > 1) = 1 - P ( X <=1) = 1 -0.337 = 0.663