Question

The number of chocolate chips in a chocolate chip cookie is Poisson with a mean of 3.

a) What is the probability that a cookie has at most one chocolate chip in it?

b) In a bag of ten cookies, what is the probability that at most one cookie of the ten has at most one chocolate chip in it?

c) What is the probability (mass) function for the number of cookies that would have to be drawn until one gets a cookie with at most one chocolate chip in it?

d) A bag of cookies has 2 cookies with at most one chocolate chip and 8 cookies with more than one chocolate chip. If Sarah and Sam split the bag evenly (5 cookies each) at random, what is the probability that Sarah gets both cookies with at most one chocolate chip?

Answer 1

Let X is number of chocolate chips so

X is Poisson(3)

So

a)

We have to find P(X<1)

Now

b)

Let Y is number of chocolate cookies out of 10 choclate cookies have at most 1 chocolate chips.

Since from (a) we know that P(chocolate cookies have at most one chocolate chips)=P=0.20

Hence Y is bin(10,0.2)

So

now we have to find P(Y<1)

Calculation

c)

Let W is number of cookies have to drawn untill we get a cookie with at most 1 chocolate chips

P(at most one chocolate chips)=0.20

So it's Geometric distribution with P=0.2

So

d)

We have total 10 cookies

8 cookies have more than 1 chocolate chips

2 cookies have at most 1 chocolate chips

So we can give 5 cookies in total

ways

Now we can give two cookies with at most one chocolate chips to sarah then sarah will get 3 cookies with number of chocolate chips more than 1 from 8 remaining cookies in

Ways so required probability is