A college university is 70 female A random variable of ten s

A college university is 70% female. A random variable of ten students is selected. What is the probability that exactly three of the ten students are female? What is the probability that at least 5 are males?

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 = 3 ) = ( 10 3 ) * ( 0.7^3) * ( 1 - 0.7 )^7
= 0.009
b)
P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0) +   
= ( 10 4 ) * 0.3^4 * ( 1- 0.3 ) ^6 + ( 10 3 ) * 0.3^3 * ( 1- 0.3 ) ^7 + ( 10 2 ) * 0.3^2 * ( 1- 0.3 ) ^8 + ( 10 1 ) * 0.3^1 * ( 1- 0.3 ) ^9 + ( 10 0 ) * 0.3^0 * ( 1- 0.3 ) ^10
= 0.84973

P( X > = 5 ) = 1 - P( X < 5) = 0.150268