# Practice Problem on Probability ( ECE302 )

If you are playing in a poker game, what is the probability of getting a royal flush, then getting four of a kind on the next hand using a 52 card deck?

If we analyze the cards in the deck, we can find that there are 52 cards in the deck. There are four suits of identical 13 cards.

To get a royal flush, you need to have an Ace, king, queen, jack and ten of the SAME suit, not different ones. so the probably is 20/52(any card between ten and ace) times 4/51 times 3/50 times 2/49 times 1/48 (4,3,2,1 are cards left in sequence, the deck also gets smaller) times 4 ( suits) which would be 6.156e-4 percent of the time.

Then to get a four of a kind, you would need 4 of any card with different suits. So you would take 1 ( any card) times 3/51 times 2/50 times 1/49. That is .0048 percent of the time.

Multiplying them together, you get 2.956e-6 percent chance of this happening. Good luck with that!

I'm a little rusty on my cards understanding, so hopefully I got the numbers in the right order.

## Comments

- Congratulations on being the first one to dare posting something on this topic! -pm
- Where does the concept of conditional probability come up?
- Answer here.

- I think independence also comes up in this problem.
- Note: if I were to ask this on an exam, I would define what is a royal flush. So don't worry if you don't know all the poker terms. -pm