Jeremy sells a magazine which is produced in order to raise

Jeremy sells a magazine which is produced in order to raise money for homeless people. The probability of making a sale is, independently, 0.20 for each person he approaches. Given that he approaches 15 people, find probability that he will make: a.) 2 or fewer sales b.) exactly 4 sales c.) more than 3 sales

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 < = 2) = P(X=2) + P(X=1) + P(X=0)   
= ( 15 2 ) * 0.2^2 * ( 1- 0.2 ) ^13 + ( 15 1 ) * 0.2^1 * ( 1- 0.2 ) ^14 + ( 15 0 ) * 0.2^0 * ( 1- 0.2 ) ^15
= 0.398

b)
P( X = 4 ) = ( 15 4 ) * ( 0.2^4) * ( 1 - 0.2 )^11
= 0.1876

c)
P( X < = 3) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 15 3 ) * 0.2^3 * ( 1- 0.2 ) ^12 + ( 15 2 ) * 0.2^2 * ( 1- 0.2 ) ^13 + ( 15 1 ) * 0.2^1 * ( 1- 0.2 ) ^14 + ( 15 0 ) * 0.2^0 * ( 1- 0.2 ) ^15
= 0.648
P( X > 3) = 1 - P ( X <=3) = 1 -0.6482 = 0.3518