In deciding whether to call a bet, skilled hold’em players often count their outs – the cards that will complete a drawing hand. That’s important!
A group of us were discussing how we use our outs. The question came up: Should we include the outs for cards that are no longer available? Obviously, unavailable outs cannot help make your hand.
In a game of $4-$8 limit hold’em with a full table, you and eight opponents vie for the pot. On the turn, you have A-9 spades in the hole, and 10-5 spades are on the board, accounting for four. There are nine more spades remaining (13-4). These are your “virtual” outs, any one of which will give you the nut flush.
On the turn, multiply your 9 outs by 2.2 for approx. 20% probability. Then, the card odds are simply (100-20) ÷ 20 or approximately 4-to-1 against making your flush on the River. (This information is accessible in charts published in many sources, making it easy to translate your outs into card odds.) Glancing at the pot shows the pot odds are higher, so calling a bet to see the River is a wise decision.
With your two holecards (two spades) and the four cards on the board (including two more spades), you actually have seen just six. There remain two groups of unseen cards, some of which are no longer available:
There are 16 unseen holecards held by your eight opponents plus the three burn cards, some of which may be spades – no longer available to you. That’s a total of 19 unseen cards that are not available.
The dealer holds the remainder of the deck (the stub) – 27 cards (52 – 6+19 or 25) = 27 from which the river card will be dealt. How many spades (outs) can you expect in this group – the only ones still available? These are the “real” outs, from which you can catch another spade on the River – no others.
There are nine spades (13-4) among the two groups of your unseen cards. Assuming a normal distribution, these remaining nine spades are dispersed among these two groups of unseen cards, of which there are 19+27 = 46 total, in the following manner: 100 x 19/46 = 41% (approx.); 100 x 27/46 = 59% (approx.)
Group 2 – the unseen cards still available – has 59% of the remaining nine spades; that’s five spades (approx.) available to you. Instead of the originally presumed (virtual) nine outs, the more realistic number is just five. Those are the only remaining spades that are actually available – the “real” outs.
At first glance, using the real outs – approximately five rather than nine spades – the probability (chance) of making your flush would seem to be much lower. But, with the help of Tom Green (retired college math professor and author of the Texas Hold’em Poker Textbook; www.PokerTextbook.info), we can calculate the probability in each case, from which the card odds can be determined.
Calculating the probability (chance) of getting another spade on the river:
• For the nine virtual outs: (100/46) x 9, where 46 is the number of all unseen cards.
• For the five real outs: (100/27) x 5, where 27 is the number of available unseen cards.
Doing the math, if we did not round off (approximate) in our calculations, and used the precise values, both probabilities would be identical!
Bottom Line: You can continue to use the “old” method with the virtual outs; it’s much easier. And, you can comfortably use one of the many charts published in various sources to convert the number of outs to the card odds.
Acknowledgement: In preparing this column, in addition to Tom Green, I received much valuable support from several outstanding poker experts: Dr. Alan Schoonmaker, Linda Johnson, Jan Fisher, Jonathan Little, Ron Ross, and Robbie Strazynski.