Question

Perhaps the most common error in ranking poker hands involves confusing the order of a straight and a flush.

A straight is 5 cards with consecutive rank, like 45678. The cards rank A,2,3,4,5,6,7,8,9,10,J,Q,K,A. Note that an ace ("A") can be used as high (as in 10JQKA) or as low (as in A2345).

A flush is 5 cards of the same suit.

Compare the probability of getting a straight with that of getting a flush. Which hand is more likely?

Answer 1

There are 52C5 ways to pick 5 cards from a deck of 52 (or 52! divided by 47! and divided by 5!), but that is a denominator which we ultimately won't need, unless you want the explicit probabilities as opposed to just the relative likelihood. Now how many ways can we make a straight, and how many ways can we make a flush?

The flush is actually simpler. First, let's ask the question of the number of ways to make a flush of hearts, then multiply by 4. To pick 5 cards which are all hearts, you just need to pick 5 out of 13 or 13C5 (13! divided by 8! divided by 5!).

Answer 2

To count the number of straights, we start with the number of straight ranks, namely, a 5 high straight, a six high straight, etc up to an Ace high straight. How many ranks is that?

Next for each rank, how many ways are there to get that? Well there are four possible cards which could be drawn for each position, so we need to multiply by 4^5.

Finally, because we know there is a special hand which is not exactly a "straight" or a "flush" because it is a "straight flush". There are very few of these, but they will need to be subtracted from both your count of straights and your count of flushes.

Answer 3

Also, I would argue that the most common error in ranking poker hands comes when someone has two pair versus an opponent with an overpair (or sometimes top pair), but the board pairs without giving the first player a boat.