Originally Posted by Hello Kitty
If there has already been recent threads about 5-card draw probabilities or easily available and useful websites, I apologize in advance.
Suppose in basic 5-card draw, we have 99xyz and drew 3.
1. What are the odds of improving to two pairs?
2. What are the odds of improving to trips?
3. What are the odds of improving to a full house?
4. What are the odds of improving to quads?
And finally, would "two pairs or better" mean the sum of these percentages, or is that calculated in some other sort of way?
Suppose you held onto a kicker, say 99Axy and drew 2 instead. What are your odds of improving to two pairs or better?
Finally, my last question is if you hold AKxzy, would it be better to throw away the King and draw 4 or to keep the King and draw 3 to improve to one pair or better? I know in video poker, we would often hold onto a King especially if it is suited because of the bonuses, but that doesn't necessarily mean it improves the probability of improving your hand.
I'm sure these odds or probabilities can be found at several
sites; however, in the case of playing in a poker game where
there has already been some folding, certain cards, notably
aces and kings are more likely than if you were just playing
heads up (since almost all players will play KK or AA in a
five- or six-handed game). Assuming each unseen card has
equal probability of being drawn, here are the probabilities.
DRAWING THREE TO A PAIR
Combinations are out of C(47,3) = 16 215. C(n,r) denotes
the number combinations of n objects taken r at a time =
n!/[ (n-r)! r! ].
1. Two pairs: 9C(4,2)(44-3)+3C(3,2)(44-2) = 2 592
or probability of about 0.1598519889 or odds of about
5.255787037 to 1 against.
2. Trips: 2[C(9,2)x4x4+9x4x3x3+C(3,2)x3x3] = 1 854
or probability of about 0.11433857540 or odds of about
7.7459546926 to 1 against.
3. Full house: 2[9C(4,2)+3C(3,2)]+9C(4,3)+3C(3,3) = 165
or probability of about 0.01017576318 or odds of about
97.27272727 to 1 against.
4. Quads: C(2,2)(48-3) = 45
or probability of about 0.002775208141 or odds of about
359.3333333 to 1 against.
Total combinations above: 2 592 + 1 854 + 165 + 45 = 4 656
One can then calculate the "two pairs or better" chances by
summing the above, but if the above isn't computed, it's
easier to find the probability of not improving and subtract
5. No improvement: C(9,3)x4x4x4+C(9,2)x4x4xC(3,1)x3+
9x4xC(3,2)x3x3+C(3,3)x3x3x3 = 11 559 or the probability is
about 0.7128584644; therefore, the probability of improving
to two pairs or better is 0.2871415356. As a check, adding
the combinations: 4 656 + 11 559 = 16215 = C(47,3).
DRAWING TWO TO A PAIR AND A KICKER
As mentioned in the first paragraph, in the context of a
game, if there are several folds, it will be slightly more likely
as compared to heads up play to catch an ace.
Combinations are now out of C(47,2) = 1 081.
1. Two pairs: 3[44-2]+9C(4,2)+2C(3,2) = 186
or probability of about 0.1720629047 or odds of about
4.811827957 to 1 against.
2. Trips: 2[44-2] = 84
or probability of about 0.07770582794 or odds of about
11.86904762 to 1 against.
3. Full house: 2x3+C(3,2) = 9
or probability of about 0.008325644218 or odds of about
119.1111111 to 1 against.
4. Quads: 1
or probability of about 0.0009250693802 or odds of exactly
1080 to 1 against.
Again, one can simply add the above combinations for
improving to two pairs or better: 186 + 84 + 9 + 1 = 280.
Alternatively, without the calculations above, one can
calculate the probability of not improving:
5. No improvement: C(9,2)x4x4+9x4x2x3+3x3 = 801
or probability of about 0.7409805735; therefore, the
probability of improvement is about 0.2590194265 or about
2.860714286 to 1 against.
DRAWING TO AK
Again, as mentioned in the first paragraph, if playing in a
game with some folds, aces and kings are bit more "live" as
compared to heads up play. You normally prefer to draw
three to AK instead of four to an ace since a pair of kings
has a decent chance of winning the pot even though
technically, drawing four to an ace increases the chances
of making a pair; however, if you were the SB or BB, AK
could be the best hand predraw so discarding the king could
cost you the pot.
I've made these calculations before, but with AK(offsuit)
and no card in the ten to queen range, there are 7 772
combinations of improvement to a pair or better. Then,
there are the following "adjustments" to the combinations:
one card in Q/J/T range: -16
two cards in Q/J/T range: -28
AK "onsuit" without third card of suit: +164
AK "onsuit" with third card of suit: +119
Then, there is another "minor adjustment" of +1 if you toss
out a "suited ten" from AKT(suited); presumably, you would
often draw two to AKQ(suited) and AKJ(suited) as a pair of
jacks often wins in a draw game scenario.
Thus, the probability of improving on AK depends on the
cards in the ten to queen range and being "onsuit" and if my
calculations are correct, the average adjustment is very
roughly about +23.1 so the probability is about 0.4807 in
improving to a pair or better on average
In the best case, the probability of improvement to a pair or
better is (7 772 + 164)/16 215 or about 0.4894233734 or
DRAWING FOUR TO AN ACE
Calculations are bit more difficult as there are now C(47,4) =
178 365 combinations. Making a flush now depends on how
many cards of the same suit are discarded (could be zero,
one or two) and making a straight depends on how many
cards in the ten to king and deuce to five range are discarded.
Even in the worst cases, drawing four to an ace will improve
to a pair or better more than 1/2 of the time; however, that
is not to say that it is clearly better in the context of a draw
poker game: e.g., if you are the big blind and are heads up
with a small blind that limps, drawing three can give you a
decent chance of bluffing whereas by drawing four, bluffing
won't nearly be as effective against typical opponents.
Here's a nearly "worst case" hand: A
Combinations out of 178 365.
AA: 3[C(8,3)x64+C(8,2)x16x12+8x4xC(4,2)x9+C(4,3)x27] =
unseen rank: 8C(4,2)[C(7,2)x16+7x4x12+C(4,2)x9] = 34 848
seen rank: 4C(3,2)[C(8,2)x16+8x4x9+C(3,2)x9] = 9 156
total for one pair: 76 392
2. Two pairs:
Aces up: 3[8C(4,2)(44-4)+4C(3,2)(44-3)] = 7 236
Other: C(8,2)x6x6+8x6x4x3+C(4,2)x3x3 = 1 638
total for two pairs: 8 874
C(3,2)[C(8,2)x16+8x4x12+C(4,2)x9] + 8C(4,3)[44-4] +
4C(3,3)[44-3] = 4 102
2x4x4x3x3 = 288
5. Flush: C(10,4) = 210
6. Full house: C(3,2)[8x6+4x3]+3[8x4+4x1] = 288
7. Quads: C(3,3)[48-4]+8 = 52
Total combinations for the above: 90 206
Thus, the probability of making 22+ is 90 206/178 365 or about
0.5057382334 or almost 50.6%, but clearly above 1/2. Thus,
drawing four to an ace will be better than drawing three to AK
if the objective is to maximize the chance of making 22 or
better; however, as mentioned earlier, drawing four is usually
inferior for other reasons in an actual draw poker game. As a
clear example: you are the big blind with the above hand and
the cutoff limps and his range is predominantly TT-QQ. Now,
you simply maximize your chances of making QQ+ which turns
out to be drawing three to AK.