Building your mathematical poker game (preflop)

One of the first things my poker coach did, after looking at a sample of my hands, was to send me to hit the books and tell me not to play poker until I was done with probability homework.  He wanted me to work the math out for my game to understand the values of hands, and how they’re affected by opponents holdings. 

I used a combination of PokerStove (which my coach wrote) and the poker calculator built into Poker Academy to do some math.  My method was to tell PokerStove that my opponents were playing a specific range of hands and then see how my hands survived.  So for example, I took a hand like Tens (TT) and measured them against opponents playing a range of hands, such as: 66+,A2s+,K6s+,Q8s+,J8s+,T8s+,A7o+,K9o+,QTo+,JTo.

That notation stands for a range of hands of your opponent playing:

  • 66: Pocket pairs, 6’s or higher
  • A2s+: Suited Aces, Ace-Two or higher
  • K6s+: Suited Kings, Six kicker or higher

You get the idea.

Pokerstove simulates all the possible five card community boards, tallying who wins the hand.  Here’s the results:

22,939,695  games    12.453 secs     1,842,101  games/sec

    equity     win     tie           pots won     pots tied   
Hand 0:     61.947%      61.10%     00.85%           14016461        195527.00   { TT }
Hand 1:     38.053%      37.21%     00.85%            8534902        195529.00   { 66+, A2s+, K6s+, Q8s+, J8s+, T8s+, A7o+, K9o+, QTo+, JTo }

The tens win 61% of the time, the range of junk (25.2% of the top hands) wins 38% of the time.   The more interesting math is when you normalize this to the expected hand equity.  The 66% is simply the chance of winning the hand, but your expected value  over playing any two cards is 123%.  How did I get this?

If this was a coin flop, you and your opponent would each win 50% of the time.  Given the fact that you and your opponent will be dealt the same cards over the long haul, before the card are dealt out you and your opponent have an equal chance of winning the hand.  You have to figure that you are "entitled" to win a hand of poker against one opponent 50% of the time.  If you have a hand that will win more often than 50% of the time, then you’ve got a big edge.  How much of an edge?  Well when you divide 61.1% chance of winning by 50% chance of winning (61.1% / 50%) = 123.9%. 

That’s your expected value.  For every bet you put in, you should expect to receive 123.9% of it back in the long run, even if you lose several individual hands.

The methodology for the numbers works no matter how many opponents you have, though practically you rarely have more than 9 opponents at a poker table. 

Imagine you were playing against 2 opponents the same pair of Tens.

16,415,185  games    13.915 secs     1,179,675  games/sec

    equity     win     tie           pots won     pots tied   
Hand 0:     43.199%      42.37%     00.84%            6954514        137548.83   { TT }
Hand 1:     28.717%      27.71%     01.01%            4549167        165326.33   { 66+, A2s+, K6s+, Q8s+, J8s+, T8s+, A7o+, K9o+, QTo+, JTo }

Hand 2:     28.085%      26.93%     01.16%            4420565        190189.33   { 66+, A2s+, K6s+, Q8s+, J8s+, T8s+, A7o+, K9o+, QTo+, JTo }

On one hand you might think, "Oh crap, my hand was a 61% to win the pot, and now it’s 43% to win", but no, it’s actually better.  Because there are three of you there’s a bigger pot.  And because there are three of you, your "share" of the hand before the cards are dealt is 33%.  Suddenly your hand has a (43.199% / 33%) = 130.9% expected value.  It got stronger!

So the first thing my coach did was send me to do the math on all possible starting hands, so I could understand, at least from a mathematical point of view, what hands were likely to return more money to me than I would put into them.

Take 87 suited, for example.  Many poker players will tell you of the power of suited connecting cards, because of their ability to make multiple good hands, like flushes and straights.  The problem is that depending on your opponents (both their number and their potential range), they aren’t necessarily favorites.  Look at this analysis of 87s vs 2 players playing the top 25% of hands:

  11,430,622  games     8.365 secs     1,366,482  games/sec

    equity     win     tie           pots won     pots tied   
Hand 0:     27.241%      26.84%     00.40%            3068116         46166.50   { 87s }
Hand 1:     36.560%      35.04%     01.53%            4005131        174548.00   { 66+, A2s+, K6s+, Q8s+, J8s+, T8s+, A7o+, K9o+, QTo+, JTo }
Hand 2:     36.199%      34.66%     01.54%            3962141        176209.00   { 66+, A2s+, K6s+, Q8s+, J8s+, T8s+, A7o+, K9o+, QTo+, JTo }

Those suited connectors are a 27.241% chance to win, which is worse than the 33% chance to win before the cards even come out.  Assuming you started this hand with a 1/3rd chance to win before the cards were dealt, playing this hand puts you farther behind your opponents despite their broad, loose range.  The expected value of this hand against these two types of opponents is 27.241% / 33% = 82.5%.  So for every bet you put into this hand in this context, you’re only getting 82.5% of that money back in winnings. 

In other words, don’t play this hand against two opponents like this.  If you run the PokerStove numbers you’ll see it becomes dead even when you’ve got 4 or 5 opponents, and begins to gain a little edge at 6 or more opponents of this type.

So my homework, which I was told to finish before I played any more poker, required me to calculate for every possible starting hand, against up to 8 opponents, the expected value of every hand in every situation.  I think in a future post I’ll share that math with you, and talk about how it’s useful in figuring out how to play a hand.