# Probabilities of Making a Flush

There are so many aspects in becoming a winning poker player. You must have the ability to make reads, play your cards and recognize patterns in opponents. We all know this. But sometimes we tend to overlook the math involved in poker. I know, I know, nobody likes math. But it is essential in understanding what correct calls to make. So from time to time, I will devote a column on probabilities of making certain hands.

Today’s topic is going to be calculating the odds of completing a flush draw in Holdem. Let’s say you look at your cards and you have 2 suited cards in your hand, for this example, say diamonds. On the flop, 2 diamonds, with no pairs hit the board. So obviously, in order for you to make your flush you need a diamond to fall on either the turn or the river. Of course the preference is always the river but let’s not get into subjects we truly can’t control. You might make the diamond on the turn, you might make the diamond on the river or you might hit a diamond on both streets. We’re going to look at all 3 scenarios, but first let’s examine the probability of a diamond hitting on both the turn and the river. You have seen the 2 cards in your hand and the 3 cards on the flop so of the 52 cards in the deck there are 47 cards remaining that you have not seen. Of those 47 cards, 9 are diamonds. In mathematical terms, there is a 9/47 chance of you hitting a diamond on the turn. Let’s go a step further. Once the turn card hits there are now 46 cards in the deck that you have not seen and if you did hit your diamond on the turn there are now 8 diamonds remaining in the deck. There is now an 8/46 chance of another diamond hitting on the river. So basically what this all means is that the chance of hitting 2 more diamonds in a row to your flush draw is 9/47 times 8/46. Go grab a calculator if you want but the calculation comes to 72/2162 which means that 72 out of every 2,162 times there is going to be one extra flush card over your made flush, approximately 1 out of 30 times.

However, most of the time when you do make your flush, the flush card is only going to hit on one of the streets, either the turn or the river. Now that we’ve seen the calculation on both hitting, it is easier for us to break it apart to either the turn or the river. As we saw in the previous calculation the probability of hitting your flush card on the turn is 9/47. If you hit your flush card on the turn the probability that there will not be another flush card on the river is now 38/46. Its 38/46 because there are 38 cards out of the remaining 46 unseen cards that are not diamonds. So this calculation is 9/47 times 38/46 and the result is 342/2162 which means that 342 out of every 2,162 times you will make your flush on the turn. The equates into approximately once out of every 6.2 times. Are you still with me here?

The last way for you to make your flush is to miss it on the turn and make it on the river. Is there a better feeling than hitting your nut flush on the river? Before the turn, 38 of the remaining 47 cards that you have not seen are not diamonds so the probability of a diamond NOT hitting on the turn is 38/47. When you miss your flush on the turn, there are now 9 diamonds remaining of the 46 cards that you have not seen so the probability of you hitting your flush on the river card is 9/46. So this calculation is 38/47 times 9/46 and the result is 342/2162 which means the 342 out of every 2,162 times you will make your flush on the river. This too is also once out of every 6.2 times. When we add both these probabilities; 72 out of 2,162 times that you’ll hit the flush card on both streets, 342 out of 2,162 times that you’ll hit the flush on the turn and 342 out of 2,162 times that you’ll hit the flush on the river, the odds of making the flush are 756 out of 2,162. This means that you should hit your flush a little better than 1 out of 3 times if you see both the turn and the river card.

As you can see, when you have 4 cards to the flush after the flop, you are usually in pretty good shape. I wouldn’t recommend pushing all your chips to the center of the table of course. There are always other factors to keep in mind. The number of players you’re up against, is there a possibility of someone beating you either with a full house or a higher flush? As you can see, math is important but you can not use it as your only guide. You must maximize your chances of winning by knowing some mathematical principles as well as being aware of the situation.

Until next time, may the chips fall your way.