Aytan Eldarova is a British poker player with a background in mathematics and technical sciences. In addition to her love of poker - she won the Womens Event at the 2024 British Poker Series - Eldarova is also a keen chess player.
Many of us regulars at the poker table are familiar with the excitement of seeing a flop reveal two cards of the same suit that you’re holding, and the anticipation of winning a big pot by hitting our flush draw.
We are just as familiar with that excitement leading us into a black hole of continuing in the hand, only to see our hopes fade as the turn and river completely miss us.
How often have you played this hand? How often did you fold? And most importantly, why did you opt for one over the other? Was it outs, odds, effective stack, ranging, bet-sizing, the tournament stage, your read on the opponent?
If it is all of the above, you are correct, but what if there is one more tool that can give you an edge over your opponents and - unlike some of the above calculations - is quick and easy to execute at the table?
I call it the ‘minimum chance of the flush draw’, and it can inform this key decision by clarifying whether you actually have a 19.1% chance of making a flush on the turn, or if there is an additional factor that is setting you up to fail.
The ‘varying guaranteed minimum’
Let’s say you’re holding two cards of the same suit, say two hearts, and the flop brings two hearts and a spade.
With nine outs, you have a 19.1% chance (4.22-to-1 odds against) of making your flush on the turn. If the turn misses, you then have a 19.6% chance (4.11-to-1 odds against) of hitting the flush on the river.
If you face a small post-flop bet it is easy to call, but what if you are looking at significant bets on the flop and turn, indicating that you will be looking at an all-in on the river? That can be a tougher decision which may result in the end of your tournament life - or a huge chip lead.
But what if I told you that your 19.1% chance comes with a varying guaranteed minimum, which depends on one simple factor: the number of players at your table (and therefore the number of cards dealt, and remaining in the deck)?
Final table implications
First, let’s make sure we understand the logic of the 19.1 % chance of hitting your flush draw. There are 52 cards in the deck; Player A has two cards (both hearts) plus three on the flop (two hearts and a spade), meaning that five cards are known to Player A, 4 of which are hearts. So, we have 47 unknown cards with potentially 9 outs. Our probability is 9/47 = 0.191, or 19.1%.
Now, the question is: is this number affected by the number of players at the table, and should the Hero act differently? It may run counter to traditionally accepted wisdom, but the answer is ‘yes’.
In Texas hold’em a player can only hold two cards, thus there is a maximum number of hearts that any player can claim from the deck.
Lets compare the dynamics of 8-player and 2-player tables, when the Hero has two hearts and two come on the flop.
On an 8-player table, 7 opponents between them have been dealt a total of 14 cards, and so they may potentially now hold all 9 hearts. There is a possibility that no hearts remain in the deck so, depending on the cards held by your opponents, your chances of making the flush on the turn range from 0% to 19.1%.
On a 2-player table, the Villain has been dealt 2 cards, so the number of hearts they can hold can only be 0, 1 or 2. So, we now have a certainty that the deck of cards holds at least 7 hearts. The number of cards remaining in the deck is 45, so our calculation looks like this: 7/45 = 0.156 or 15.6%. Let’s call this your ‘minimum chance’. Again, depending on the cards held by your opponent, your chances of making the flush on the turn now range from 15.6% to 19.1%.
Below you can see how the minimum chance changes, based on the number of players at the table:
- 9-player table 0/31 = 0%
- 8-player table 0/33 = 0%
- 7-player table 0/35 = 0%
- 6-player table 0/37 = 0%
- 5-player table 1/39 = 2.6%
- 4-player table 3/41 = 7.3%
- 3-player table 5/43 = 11.6%
- 2-player table 7/45 = 15.6%
Is this critical knowledge that could impact the way you play? Possibly, possibly not. Is it an interesting way of looking at draws in poker that’s not been identified before? I think so.
The mathematical engine of poker is a fascinating topic for many of us - it’s rare you look under the hood and spot something new.
Dara O’Kearney responds
For a second opinion on this intriguing take, PokerOrg reached out to Dara O’Kearney, a strategy guru with an impeccable grasp of the independent chip model (ICM) and the mathematics of equity in poker.
Over to Dara…
This is a very interesting perspective I haven't seen before, and here's a hypothetical example to illustrate: a target satellite, two-player table.
The target is 100K; we start the hand with 90K behind. Our opponent, playing a short stack, raises and we defend a suited hand in the big blind.
We flop the nut flush draw. We check and face an all-in bet. If we assume the following...
- if we make the flush, we always win
- if we don’t, we always lose
...then the conventional approach is to assume our actual equity is 34.96% - (1-(38/47) *(37/46)) - and call if the ICM-adjusted equity we need to call is 34.96% or less.
However, under the new theory, our chance of hitting the flush is one of the following:
- (1-(36/45)* (35/44)) or roughly 36.36% when opponent holds none of our suit (which happens roughly 56% of the time)
- (1-(37/45)*(36/44)) or roughly 32.73% when opponent holds one of our suit (which happens roughly 38% of the time)
- (1-(38/45)*(37/44)) or roughly 28.9% when opponent holds two of our suit (which happens roughly 6% of the time)
Although this averages out to 34.96% and A is the most common scenario, it changes our calculation. Having less equity than we need costs us more than having more than we need benefits us.
In this case, we are only guaranteed to have at least 28.9% and now the ICM-adjusted equity we need to call is somewhere between 28.9% and 34.96%.
I chose this example because it is the simplest to calculate but the same considerations apply in any ICM situation.
What do you think? Share any thoughts in the comments below.
You can get more coaching and study materials from Dara O'Kearney at Simplify Poker. His 'Poker Solved' series of books, written with Barry Carter, is available at Amazon.
Images courtesy of Danny Maxwell Photography/Eloy Cabas/Rational Intellectual Holdings Ltd.
