The Expected Value of Chasing a Flush when Heads Up

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Many books and articles discussing “Expected Value” and pot odds use a flush as an example of how to determine whether or not you should enter a pot, and with how much strength. It is generally accepted that the odds are in your favour when you have four suited cards post-flop with two or more players going against you, but not always advisable when there is only one opponent against you (as the odds of getting the flush alone are not in your favour). However, when you have two high ranking suited cards in your hand, you have other alternatives that you may also want to consider.

The expected value figure for catching the flush is 39% and is calculated by working out that there are nine unseen cards of your suit in the unseen pack of 47 (“unseen” as they are cards that you cannot se rather than anybody else). To calculate the odds of the fifth suited card being drawn on the turn, you divide 9 by 47 (0.19 or 19%), and if it does not show, the odds of it appearing on the river are then 9 divided by 46 (the remaining unseen cards) which is 0.20 or 20%. By adding the two factors together is how the figure of 39% is arrived at.

Assuming that one of the suited cards in your hand is an ace, you also have the chance of getting top pair to add to the probability of successfully getting the flush. The odds of this happening are worked out by dividing the three unseen aces (assuming one was not flopped) by 47 and 46 respectively to give a total of 13%. If you add this figure to the previously derived 39%, the total probability of catching the flush or top pair is 52% – and marginally in your favour against just one other opponent.

One further point about chasing a flush is when you do not hold top flush and, although you have drawn it on the turn, what the chances are of your playing colleague(s) having a higher card than you either already, or should a further suited card be drawn on the river. To calculate the odds that the other player may be holding a higher flush than you on the turn, you know that there are two suited cards in your hand, two in his and three (from four dealt cards) on the table. This means that in order for him to have any two cards of the same suit, there are seven unseen (say) hearts in a total of 46 unseen cards – or 15%. Then you divide the number of cards higher than your highest ranking heart by 13, and multiply the two answers together. For example, if you had a nine high flush, there are five cards of higher rank (you also deduct any showing in the flop or turn) so you would divide 5 by 13 (38%) and multiply it by 15% = 0.06 or 6%. Therefore there is a 6% (or 16/1) chance that not only does the player in the heads up situation hold two hearts, but his top catd is higher than yours.

To calculate the chance that he holds the ace of hearts or other higher card when a further suited card turns up on the river, simply divide the number of higher hearts that can beat your top card by the total number remaining in the pack. Therefore if you have a nine high flush, the odds of a further heart being drawn on the river and your opponent holding either T/J/Q/K/A of that suit, are 5 divided by 45 (11%) or 1 in 9. The higher your top card is, the longer the odds of his chance of winning.

Of course, there are a number of alternative potential outs for both you and your adversary, and if the other player is in front of you in the betting, you will be able to gauge the potential strength of their hand by the manner of their betting. However, if you take advantage of the combined odds being in your favour, you will win more often than not.

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