Probability of Choosing a person that likes cake.

Question

You have $20$ people in a population. $5$ people prefer cupcakes over cake and $15$ prefer cake over cupcakes. you will choose $5$ people at random. What is the probability that you will choose $2$ people that prefer cupcakes and $3$ people that prefer cake?

Answer 1

The correct answer is $$\frac{\binom{5}{2}\binom{15}{3}}{\binom{20}{5}}.$$

There are $\binom{5}{2}$ to choose the people who like cupcakes and $\binom{15}{3}$ to choose the people who like cake. Overall, you are picking $5$ people from $20$ people which is $\binom{20}{5}.$

Answer 2

The probability is $$\frac{\binom{5}{2}\binom{15}{3}}{\binom{20}{5}}.$$

You should read about the Hypergeometric Distribution if you want to learn more.

Answer 3

First, you need to calculate how many ways to pick $5$ people out of $20$:

$$\binom{20}{5} = \frac{20!}{5! (20-5)!} = 15504$$

Then $\mid \Omega \mid = 15504$, those are all the possible ways to pick $5$ people out of $20$.

So, you need to calculate how many ways to pick $2$ out of $5$ (those are the people who prefer cupcakes):

$$\binom{5}{2} = 10$$

Finally, how many ways to pick $3$ out of $15$ (people who prefe cake):

$$\binom{15}{3} = 455$$

So, your favorable cases are $455 \times 10 = 4550$, and using classic probability (favorable cases over total cases):

$$\frac{\binom{15}{3} \binom{5}{2}}{\binom{20}{5}} = \frac{4550}{15504} = 0.2934$$

The probability you're looking for is $0.2934$.