Question

Lemonbee’s Restaurant will serve a free dessert to customer’s on his/her birthday, provided that the person has registered for Lemonbee’s email list. Currently, there are 500 people who are on the email list and 4 of them have their birthday today. Lemonbee’s has 20 customers today. Assume that every customer who celebrates a birthday with free dessert is on Lemonbee’s email list.

a) Define X as the number of today’s customers at Lemonbee’s who have a birthday today. What is the distribution, parameter(s), and support of X?

b) What is the probability that at least one of the current customers has a birthday today?

c) It costs Lemonbee’s $2.35 to provide the free dessert. What is the expected value and standard deviation of the cost for Lemonbee’s free birthday desserts today.

d) Is there a valid approximation that can be used to calculate probabilities related to the number of today’s customers with birthdays today? If so, state the distribution, parameter(s) and support, along with the reason that the approximation is valid. If not, explain why not.

e) If you stated that there is a valid approximation in part e) use it to find the probability that 2 or 3 of today’s customers have their birthday today. If not, find the exact probability that 2 or 3 of today’s customers have their birthday today.

Answer 1

a)

Here X has hypergeomteric distribution with parameters as follows:

Population mean: N= 500

Number of successes (number of people have birth today) in population: M = 4

Sample size: n = 20

The number of successes (number of people have birth today) in sample: x (here x can take values 0, 1, 2, 3, 4)

The pdf of X is

b)

The probability that at least one of the current customers has a birthday today is

Answer: 0.1511

c)

First we need to find the mean and standard deviation of X. So

Let Y shows the cost for Lemonbee’s free birthday desserts today. We have

Y = 2.35*X

So,

Answers: the expected value and standard deviation of the cost for Lemonbee’s free birthday desserts today is $0.376 , $0.9183 respectively.

d)

Since

So binomial approximation can be used.

Using approximation, X has binomial distribution with parameters as follows:

n=20, p = 4/500 = 0.008

The pdf of X is

e)

Answer: 0.0110