What is the probability that your friend chooses chocolate and vanilla when selecting two different flavors of ice cream at a store?

Question

A local grocery store offers $30$ different flavors of ice cream. You ask your friend to stop and get two different pints of ice cream. What is the probability of your friend selecting chocolate and​ vanilla?

Answer 1

In the following, we assume that your friend is equally likely to choose any of the flavors.

Method 1: Order of selection does not matter.

The number of ways of selecting a subset of $k$ elements from a set with $n$ elements (the number of ways of making an unordered selection) is $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$

In how many ways can your friend choose two of thirty flavors?

In how many ways can your friend choose two flavors selected from chocolate and vanilla?

Divide the number of ways your friend can choose both chocolate and vanilla by the number of ways your friend can choose two of the thirty flavors.

Method 2: Order of selection matters.

The favorable cases are your friend chooses chocolate then chooses vanilla or your friend chooses vanilla then chooses chocolate.

What is the probability that your friend chooses chocolate? What is the probability that your friend then chooses vanilla from the remaining flavors?

What is the probability that your friend chooses vanilla? What is the probability that your friend then chooses chocolate from the remaining flavors?

Since these sequences of events are mutually exclusive, you can find the probability that your friend chooses chocolate and vanilla by adding the probabilities of the two events above.