If n5 and 030 what is the probability of the following a X4

If n=5 and =0.30, what is the probability of the following?

(a)

X=4

(b)

X3

(c)

X<2

(d)

X>1

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 ) * ( 0.3^4) * ( 1 - 0.3 )^1
= 0.0284
b)
P( X < = 3) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 5 3 ) * 0.3^3 * ( 1- 0.3 ) ^2 + ( 5 2 ) * 0.3^2 * ( 1- 0.3 ) ^3 + ( 5 1 ) * 0.3^1 * ( 1- 0.3 ) ^4 + ( 5 0 ) * 0.3^0 * ( 1- 0.3 ) ^5
= 0.9692
c)
P( X < 2) = P(X=1) + P(X=0)
= ( 5 1 ) * 0.3^1 * ( 1- 0.3 ) ^4 + ( 5 0 ) * 0.3^0 * ( 1- 0.3 ) ^5   
= 0.5282

d)
P( X < = 1) = P(X=1) + P(X=0)   
= ( 5 1 ) * 0.3^1 * ( 1- 0.3 ) ^4 + ( 5 0 ) * 0.3^0 * ( 1- 0.3 ) ^5
= 0.5282
P( X > 1) = 1 - P ( X <=1) = 1 -0.5282 = 0.4718