Calculating keno odds?

Question

In keno, the casino picks 20 balls from a set of 80 numbered 1 to 80. Before the draw is over, you are allowed to choose 10 balls. What is the probability that 5 of the balls you choose will be in the 20 balls selected by the casino?

My attempt: The total number of combinations for the 20 balls is $80\choose20$. However, I get stuck at the numerator. I thought it will be $\binom{80}{10}\binom{10}5$ but that's wrong.

Thanks.

Answer 1

Another way to think about this is to realize that the casino must choose $5$ balls from the $10$ that you chose and $15$ balls from the $70$ that you didn't choose. So: $$P = \frac{\binom{10}{5} \binom{70}{15}}{80\choose 20} \approx 0.0514...$$

Answer 2

Without loss of generality, assume the casino picks balls 1 to 20. Then for the stated scenario to happen:

• Five of your picks are within $[1,20]$: $\binom{20}5$ ways
• The other five are within $[21,80]$: $\binom{60}5$ ways

There are $\binom{80}{10}$ picks altogether, so the probability that five balls match is $$\frac{\binom{20}5\binom{60}5}{\binom{80}{10}}=0.0514\dots$$