4 player flush draw odds

Question:  In a four player game, how often will you make a flush when setting 2-4 flush cards in a hand?

With four players, we start seeing almost all of the deck in the late draws.  This means that there are some situations that are impossible, like having four flush cards in a hand with no other flush cards out and 1 draw to go.  In this case, there are 4 cards left to draw in the deck, which, even if they were all flush cards, would mean the deck had only 8 cards in that suit.  These situations are indicated in the table with "IMP".

There are also situations where the probability of getting a flush goes up as the draws progress, as with setting two cards and having no dead cards in any of the other three hands.  With this few dead cards out, the deck in the late draws is almost all flush cards.  Of course, these situations will be relatively rare.

Chance of completing a flush setting 4 flush cards, 4 player model 

dead cards	Draw 1	Draw 2	Draw 3	Draw 4	Draw 5	Draw 6	Draw 7	Last Draw
0	95.3%	95.7%	96.3%	97.0%	98.1%	99.5%	IMP	IMP
1	93.0%	93.4%	94.2%	94.8%	96.2%	98.2%	100.0%	IMP
2	89.7%	90.2%	90.7%	91.8%	93.0%	95.5%	100.0%	IMP
3	84.9%	85.7%	86.2%	87.0%	88.5%	90.8%	96.5%	IMP
4	78.7%	79.1%	80.0%	80.4%	81.8%	84.0%	89.3%	IMP
5	70.4%	70.6%	71.3%	71.9%	72.9%	74.3%	78.4%	100.0%
6	59.0%	59.3%	59.6%	60.3%	60.9%	61.7%	64.2%	75.1%
7	44.2%	44.3%	44.6%	44.8%	45.1%	45.3%	46.3%	50.1%

Chance of completing a flush setting 3 flush cards, 4 player model 

dead cards	Draw 1	Draw 2	Draw 3	Draw 4	Draw 5	Draw 6	Draw 7	Last Draw
0	80.7%	81.7%	83.0%	84.8%	88.0%	95.5%	IMP	IMP
1	74.2%	75.1%	76.1%	77.8%	80.9%	87.2%	IMP	IMP
2	66.8%	67.3%	68.1%	69.4%	71.7%	76.1%	100.0%	IMP
3	57.7%	58.2%	58.5%	59.7%	60.7%	63.5%	75.1%	IMP
4	47.4%	47.8%	48.0%	48.7%	49.0%	49.9%	53.8%	IMP
5	36.7%	36.6%	36.7%	36.5%	36.6%	36.3%	35.5%	IMP
6	25.3%	25.2%	25.2%	24.8%	24.4%	23.6%	21.5%	0.0%
7	14.7%	14.5%	14.5%	14.0%	13.4%	12.7%	10.8%	0.0%

Chance of completing a flush setting 2 cards, 4 player model

dead cards	Draw 1	Draw 2	Draw 3	Draw 4	Draw 5	Draw 6	Draw 7	Last Draw
0	57.4%	58.1%	58.9%	60.4%	63.3%	75.0%	IMP	IMP
1	48.8%	48.7%	49.3%	50.1%	51.0%	54.5%	IMP	IMP
2	39.6%	39.8%	39.6%	39.5%	39.2%	38.2%	IMP	IMP
3	30.9%	30.7%	30.2%	29.4%	28.1%	25.4%	0.0%	IMP
4	22.6%	22.3%	21.5%	20.7%	19.2%	16.0%	0.0%	IMP
5	15.0%	14.5%	13.9%	13.1%	11.8%	9.2%	0.0%	IMP
6	8.4%	8.3%	7.9%	7.4%	6.3%	4.5%	0.0%	IMP
7	4.0%	3.7%	3.6%	3.2%	2.7%	1.8%	0.0%	0.0%

What's this?

This blog is a notebook to keep results of a set of Open Face Chinese Poker simulations aimed at finding a basic set of probabilities for some common play situations.  Because the probabilities are determined by repeated random simulation, all probabilities are estimates.

If I get a little more ambitious, I may derive the closed form of these, which will be more precise.

Heads-Up Flush Draw Odds

Question #1:  Playing heads up, if you set 2-4 flush cards, how often will you complete a flush?  

The following chart gives the odds (in heads-up open face) of making a flush draw given how many flush cards you have set, how many flush cards are in the other player's hand, and how many draws remain.

The first column of the table gives the odds of completing the flush by the end of the hand if you have 8 draws left (that is, immediately after the set), then 7 draws, etc., until the last column, which is the odds with a single card left to draw and set.  So, for example, if you have set three hearts in the back hand, and your opponent has two hearts, and there are six draws left, you have a 37.0% chance of making the flush by the end of the hand.

Chance of completing a flush with 4 suited cards set, 2 players

Dead cards	Draw 1	Draw 2	Draw 3	Draw 4	Draw 5	Draw 6	Draw 7	Last Draw
0	88.2%	85.9%	82.8%	78.7%	72.8%	64.3%	51.5%	32.1%
1	84.6%	82.2%	78.4%	73.9%	67.8%	59.0%	46.8%	28.4%
2	80.1%	77.1%	73.4%	68.6%	61.9%	53.7%	41.9%	24.8%
3	74.6%	71.0%	67.3%	62.1%	55.7%	47.6%	36.3%	21.5%
4	67.3%	63.8%	59.9%	54.8%	48.9%	41.2%	31.0%	17.6%
5	58.4%	55.2%	51.4%	46.6%	40.9%	33.8%	25.2%	14.1%

Chance of completing a flush with 3 suited cards set, 2 players  

Dead cards	Draw 1	Draw 2	Draw 3	Draw 4	Draw 5	Draw 6	Draw 7	Last Draw
0	62.7%	57.1%	50.7%	42.9%	33.3%	22.4%	10.4%	0.0%
1	55.6%	50.5%	44.1%	36.9%	28.2%	18.4%	8.1%	0.0%
2	48.0%	42.9%	37.0%	30.5%	22.8%	14.6%	6.5%	0.0%
3	40.4%	35.2%	30.3%	24.4%	17.9%	11.3%	4.8%	0.0%
4	32.1%	27.7%	23.3%	18.6%	13.3%	8.3%	3.5%	0.0%
5	23.8%	20.2%	17.2%	13.3%	9.5%	5.7%	2.3%	0.0%

Chance of completing a flush with 2 suited cards set, 2 players

Dead Cards	Draw 1	Draw 2	Draw 3	Draw 4	Draw 5	Draw 6	Draw 7	Last Draw
0	34.3%	28.6%	22.1%	15.3%	8.8%	3.4%	0.0%	0.0%
1	28.1%	22.7%	17.6%	11.9%	6.8%	2.5%	0.0%	0.0%
2	21.9%	17.7%	13.3%	8.8%	4.8%	1.6%	0.0%	0.0%
3	16.2%	12.7%	9.3%	6.1%	3.3%	1.2%	0.0%	0.0%
4	11.3%	8.8%	6.3%	4.0%	2.0%	0.7%	0.0%	0.0%
5	7.1%	5.4%	3.9%	2.5%	1.2%	0.4%	0.0%	0.0%