A popular coffee shop runs a contest where prize coupons are printed under the rims of its paper cups. Suppose 10% of the cups are printed with a winning coupon, and you purchase 15 cups of coffee during the promotion.

a) What type of distribution best represents this problem? Explain.

b) What is the probability that you will win at least 3 prizes?

c) What is the expected number of prizes you will win?

(a) We have a binomial distribution situation where

1. the probability of success is known (0.10) and remains constant throughout the experiement (of buying 15 cups).

2. The trials of type bernoulli (either win or lose, two possible outcomes)

3. All trials are independent of each other (assumed from context)

4. Number of trials is known and remains constant. (15)

Satisfaction of ALL of the above qualifies the situation as modelled by the binomial distribution, where the probability of x successes out of n trials each with a probability of p is given by:

P (X=x) = C (n, x) p^x (1-p) ^ (n-x)

where

n=15, p=0.10

(b) at least 3 successes

P (X>=3)

=1 - (P (X=0) + P (X=1) + P (X=2)

=1 - (0.205891+0.343152+0.266896)

=1-0.81539

=0.184061

(c) E[x]=np=15*0.10=1.5