At a local community college 57 of students who enter the co

At a local community college, 57% of students who enter the college as freshmen go on to graduate. Eight freshmen are randomly selected.

What is the probability that none of them graduates from the local college? (Round your intermediate calculations and final answer to 4 decimal places.)

What is the probability that at most seven will graduate from the local college? (Round your intermediate calculations and final answer to 4 decimal places.)

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 = 0 ) = ( 8 0 ) * ( 0.57^0) * ( 1 - 0.57 )^8
= 0.0012
b)
P( X < = 7) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 8 7 ) * 0.57^7 * ( 1- 0.57 ) ^1 + ( 8 6 ) * 0.57^6 * ( 1- 0.57 ) ^2 + ( 8 5 ) * 0.57^5 * ( 1- 0.57 ) ^3 + ( 8 4 ) * 0.57^4 * ( 1- 0.57 ) ^4 + ( 8 3 ) * 0.57^3 * ( 1- 0.57 ) ^5 + ( 8 2 ) * 0.57^2 * ( 1- 0.57 ) ^6 + ( 8 1 ) * 0.57^1 * ( 1- 0.57 ) ^7 + ( 8 0 ) * 0.57^0 * ( 1- 0.57 ) ^8
= 0.9889

c)
Mean = 8 * 0.57 = 4.56