Calculating Odds with Two Cards to Come

[ne14dirt]

Disclaimer: I’m bad at math so please bear with me. I’d also like to say that I am sure this topic has been presented before but I did a search and could not find it. When I search the internet I kept coming up with odd’s being calculated on the turn as opposed to on the flop. Now on to the question.

Please explain the math behind calculating odds with two cards to come. I’ve been using the rule of four and two as presented by Phil Gordon in his “Little Green Book” which works with “good enough” accuracy but as of a couple of hours ago it started bothering me that I can get my head around the percentages with one card to come but I can not for the life of me figure the math with two cards to come.

Example as presented in “Little Green Book”

I have 10c, 9d
My opponent has As, Kd
Flop Ac, 10d, 7s

Using the rule of four I take 5 as my number of outs times 4 = 20 or 20% that my card will hit.

But if I try and think it out, I tell myself I have 5 outs and for the discussions sake I am sure my opponent has A,K which leaves 45 cards left in the deck., five of which are my outs so I am 40/5 or 8/1. I’m guessing I’m 8 to1 to hit my card on the turn…..right. But 8-1 is only 12.5%......how do I calculate to include the river card.

[RobertJohn]

OK cutie you have 5 wins on the turn and 5 on the river.

Figure your chances of wiffing turn AND river, then take the complement of that, or, 1-(% you wiff both streets).

[ne14dirt]

Like I said, I am really bad at math. I appreciate the response but it doesn’t really get me any further. On the flop I am 40/5 or 12.5% to make my hand on the turn, on the turn I am 39/5 or 12.82% to make my hand on the river. Were does the 21% of making my hand on either the turn or river come from???

[Acevader]

As Robert John said you have to look not at your chances of hitting the turn or river but rather the probability of missing both. Thus by default if you know how often you miss both you know how often you'll hit either the turn or river.

So you say you are 12.5% to make turn and 12.82% to make river if you don't make turn. You chance of missing the turn is therefore 87.5% and river 87.18% Therefore its 1-(0.875*0.8718)*100 = 1-(0.7628)*100 or 0.2372*100

Therefore you will hit the turn or river 23.7% of the time.

[bigpooch]

The excellent book, "Ace on the River" by Barry Greenstein
explains somewhere in the Addenda how to do this in general,
but any mathematician should be able to do the same.

I'll describe how to do this in general for flop odds.

For all heads up confrontations where both hands are known
and the flop is given, there are 45 outstanding cards and
there are C(45,2) = 990 possible combinations of cards for
both the turn and the river. These are two key numbers to
bear in mind. Your chances of winning/losing/tying can be
expressed as a fraction of 990 and your equity is simply
your "wins"+"ties"/2 divided by 990.

As mentioned in another post, it's best to look at "losses"
since you are an underdog.

For the hand in question, you could "miss" in not making
two pairs, trips or a boat when the last two cards are both
not T or 9. You also should note you could back door a
straight with either J8 or 86 when you "miss". Now, there
are C(45-5,2)= C(40,2)=780 ways of getting no tens or nines
by the river, but you also win in 4x4=16 combinations for
each type of straight. Thus, you "lose" in 780-2*16=748
combinations when you "miss".

You could also "hit" and lose:
If you hit a ten, you lose when an ace appears and this
occurs in 2x2=4 combinations.

If you hit a nine, you lose when either another ace, a king,
or a 7 appears and this occurs in 3x(2+3+3)=24 combinations.

So, when you "hit", you still lose on 4+24=28 combinations.

There are no ties.

Thus, you lose on 748+28=776 combinations out of 990.

The number 990 is nice as it's close to 1000 and you simply
add about 1% to easily find percentages. You win on 214
combinations, and since there are no ties, your equity is
about 0.216.


For more than two players involved, you'll get slightly
different key numbers: 43 unseen cards and only 903
combinations. Still, once you get adept at these
calculations, many of these can be done in your head in just
a few minutes.

[ne14dirt]

I really appreciate the responses. Thank you.