In a recent study of college students 30 of all college stud

In a recent study of college students, 30% of all college students had at least one tattoo. A small private college decided to randomly and independently sample 15 of their students and ask if they have a tattoo.

Find the probability that no more than 8 students answered “Yes” to the question.

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 < = 8) = 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)
= ( 15 8 ) * 0.3^8 * ( 1- 0.3 ) ^7 + ( 15 7 ) * 0.3^7 * ( 1- 0.3 ) ^8 + ( 15 6 ) * 0.3^6 * ( 1- 0.3 ) ^9 + ( 15 5 ) * 0.3^5 * ( 1- 0.3 ) ^10 + ( 15 4 ) * 0.3^4 * ( 1- 0.3 ) ^11 + ( 15 3 ) * 0.3^3 * ( 1- 0.3 ) ^12 + ( 15 2 ) * 0.3^2 * ( 1- 0.3 ) ^13 + ( 15 1 ) * 0.3^1 * ( 1- 0.3 ) ^14 + ( 15 0 ) * 0.3^0 * ( 1- 0.3 ) ^15

= 0.9848