let x be a binomial random variable with n20 and p06 find Px

let x be a \"binomial\" random variable with n=20 and p=0.6, find P(x>9) using the normal approximation

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 < = 9) = P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 20 9 ) * 0.6^9 * ( 1- 0.6 ) ^11 + ( 20 8 ) * 0.6^8 * ( 1- 0.6 ) ^12 + ( 20 7 ) * 0.6^7 * ( 1- 0.6 ) ^13 + ( 20 6 ) * 0.6^6 * ( 1- 0.6 ) ^14 + ( 20 5 ) * 0.6^5 * ( 1- 0.6 ) ^15 + ( 20 4 ) * 0.6^4 * ( 1- 0.6 ) ^16 + ( 20 3 ) * 0.6^3 * ( 1- 0.6 ) ^17 + ( 20 2 ) * 0.6^2 * ( 1- 0.6 ) ^18 + ( 20 1 ) * 0.6^1 * ( 1- 0.6 ) ^19 + ( 20 0 ) * 0.6^0 * ( 1- 0.6 ) ^20
= 0.1275
P( X > 9) = 1 - P ( X <=9) = 1 -0.1275 = 0.8725