A department store is holding a drawing to give free shopping sprees- Probability

Question

A department store is holding a drawing to give free shopping sprees to two lucky customers. There are $18$ customers who have entered the drawing: $4$ live in the town of Gaston, $8$ live in Pike, and $6$ live in Wells. In the drawing, the first customer will be selected at random, and then the second customer will be selected at random from the remaining customers. What is the probability that both customers selected are Pike residents?

Report your answer as an exact fraction.

My answer The probability that the first customer lives in Pike is

P1=8/18=4/9

The probability that the second customer lives in Pike is

P2=7/17

The probability that both customers selected are Pike residents is

P1*P2=(4/9)*(7/17)=28/153

Am I Correct?

Answer 1

Yes, your reasoning is correct.

Alternatively we want the probability of choosing $2$ of $8$ Pike residents when choosing $2$ of $18$ total residents.

We have

$$\frac{8 \choose 2}{18 \choose 2}\approx0.183$$

Answer 2

This is a way to formalize your answer.

For $i=1,2$ let $E_i$ be the event that the $i$-th customer lives in Pike.

Then:$$P(E_1\cap E_2)=P(E_1)P(E_2\mid E_1)=\frac8{18}\frac7{17}$$

As you were told in the comments on your question we have $P(E_1)=P(E_2)=\frac8{18}$ but $P(E_2\mid E_1)=\frac7{17}$.