Wednesday, November 19 2008

Odds - Hitting the Flop

Written by FRC

Chances of hitting your flop with common starting hands.

Hitting the Flop

Knowing how likely it is that you or your opponent will hit the flop is quite important in Holdem, either to define how much you can call with a speculative hand, or to anticipate how the hands you play are likely to develop.

In the following paragraphs, we will use the math notation C(x,y) for the function returning the number of combinations of y elements in deck of size x. This is the combin() Excel function. As an example, C(52,2)=1326, which is the number of starting hands in Holdem. Feel free to skip the maths if it bother you.

Unpaired Hands

Hitting Pair or Better

Ace King, like 72 or any other unpaired hand, is 2:1 to make at least a pair on the flop.

To calculate this, we imagine a deck before dealing the flop (with 50 cards, since you only got your AK), and we strip it from the remaining aces and kings (there are 6 of them). The number of flop we can deal with this deck is C(44,3). All the flops miss your hand. The percentage of flops you won’t pair is C(44,3)/C(50,3), or about 67%, which is 2:1 once converted into odds. Note that this figure includes trips, full houses and quads (though a pair will be your most likely hand by far).

Hitting Trips

The odds of hitting trips are about 73:1.

How many flops would give trips? Those with KK or AA, obviously. Now there is just one card (any one but K or A) that can vary, so we have again 44 cards remaining. We have to multiply this by C(3,2), which is the number of way to choose two kings from the three ones in the deck. And eventually, multiply by 2 because we are considering both trip aces or trip king
This gives: 44xC(3,2)x2/C(50,3); which is about 1.35%, or 73:1.

Paired Board, No Trips

The odds for a paired board not giving us trips is 5.5:1.

This is the same kind of computation as the previous one, with 11 pairs (2 to Q) instead of just aces and kings: 46xC(4,2)x11/C(50,3)=15.5%, or 5.5:1.
Note that we don’t count trips on board either.

Hitting Two Pairs

Hitting two pairs is 49:1.

There are 3x3=9 ways of dealing AK with the 3 aces and kings left in the deck. Again, 44 cards remain for the last card of the flop: 9x44/C(50,3)=2%, or 48.5:1 (rounded to 49:1).

Suited Hands

Flopping a Flush Draw

Getting a flush draw on the flop is 8.1:1.

There are C(11,2) ways of making a draw with two cards, and there are 50-11=39 possibilities for the last card of the flop. Thus: C(11,2)x39/C(50,3)=11%, or 8.1:1. Note that we don’t count flush draws with three cards of the same suit on the flop.

Flopping a Flush Draw with a Pair

Hitting a pair along with a flush draw is 58:1.

We make the same computation as for the flush draw, except that the last card must pair us, and there are 6 cards left in the deck that do so. C(11,2)x6/C(50,3)=1.7%, or 58:1.

Pairs

Hitting a Set or Better

A pair is 7.5:1 against flopping a set or better.

The maths are easy: take all the flop without a king, this will give us the odds of failing to improve, and consequently the odds of hitting one or two kings (odds for complementary events are the same). That is: C(48,3)/C(50,3)=88%, or 7.5:1.

Ace(s) on the Flop

The odds of flopping one or more aces while failing to improve are 3.5:1 against.

There are C(50,3)-C(46,3)=4420 flops with at least an ace (the odds are 3.4:1 against, by the way). Now, there are 4 flops with an ace and giving you quads, and C(4,2)x2=12 flops with two aces and giving you a full house. This gives: 4420-4-12=4404 flops, or 3.5:1 against.

Ace(s) or King(s) on the Flop

The odds of flopping one or more aces or kings while failing to improve are 1.4:1 against.

There are now C(50,3)-C(42,3)=8120 flops with at least an ace or a king (1.4:1). We now have the same computation as in the previous paragraph, except that we multiply the figures by 2 since we must account for both aces and kings; thus, 8 flops with an ace or king ang giving you quads, and 24 flops with two aces or two kings and giving you a full house. Eventually, there are 8120-8-24=8088 unfavorable flops of that kind, which is 1.4:1 (after rounding, as usual).

More Overcards on the Flop

The odds of flopping one or more aces, kings or queens while failing to improve are 1.3:1 favorite.

The odds of flopping one or more aces, kings, queens or tens while failing to improve are 2.2:1 favorite.

This is the same type of computation as the two previous cases, so we won’t go through the maths again.

Paired Board

Flopping a paired board while failing to improve is 5.2:1 against.

We already computed this in a previous paragraph; C(4,2)x44 flops with a pair of a given rank (and not giving us a full house), that is, 264. There are 12 ranks left, thus giving 5.2:1.

Single-Suited Flops

Single-Suited Flop, No Draw

Flopping three cards of the same suit with none in our (offsuit) hand is 33:1.

This is simply C(13,3)x2/C(50,3) (since two of the four suits don’t give us a draw).

Single-Suited Flop giving a Flush Draw

Now we have one card of the same suit. This is 44:1 against.

This is now C(12,3)x2/C(50,3).

Hitting a Flush

Flopping a flush is 118:1.

This is C(11,3)/C(50,3).

Connectors

Straight or Open-Ended Straight Draw

Flopping a straight or an open-ended straight draw with:

T9 is 7.5:1
T8 is 11.8:1
T7 is 25:1

With T9, flops with 78, 8J or JQ give us a straight or an 8-way straight draw. There are 16 ways to make each of these two-card hands, thus giving: 3x16x48/C(50,3)=11.8%, or 7.5:1.
With T8, only 79 and 9J work, which gives 2x16x48/C(50,3).
With T7, only 89 works.
Note that paired board are not excluded.