Probability of Winning Fantasy 5 Lotto

Question

In Fantasy 5, you choose 5 numbers, each 1-39. You win if you choose match all 5 numbers, but you can also win if you match 4, 3, or even 2 of the numbers.

The lotto website lists the probabilities of winning all different scenarios. I can't seem to get the calculations right of winning scenario 4, 3, or 2.

5/5 numbers is just "39 Choose 5": C(39,5)=575,757.

I thought the odds to calculate 4/5 would be C(39,4)=82,251. But they say it's 3,387.

Why am I wrong?

https://www.calottery.com/play/draw-games/fantasy-5/how-to-play

Answer 1

For the 4-out-of-5 scenario there are $\binom54$ ways to choose which balls matched and $\binom{34}1$ ways to choose which balls didn't match. This gives a probability of $$\frac{\binom54\binom{34}1}{\binom{39}5}=\frac{170}{575757}=\frac1{3386.806}$$ and $\frac1{3387}$ is rounded from there. Calculations for lower-tier prizes follow:

• 3 of 5: $$\frac{\binom53\binom{34}2}{\binom{39}5}=\frac{5610}{575757}=\frac1{102.630}$$
• 2 of 5: $$\frac{\binom52\binom{34}3}{\binom{39}5}=\frac{59840}{575757}=\frac1{9.622}$$
• Any prize: $$\frac{1+170+5610+59840}{575757}=\frac{65621}{575757}=\frac1{8.774}$$