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What is the exact probability that 5 players at a table will be dealt no clubs (or hearts or whatever) ?

Happened to me when all 5 community were all clubs (2,3,5,K,9), thereby a 5 way chop.

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I think intuition tells you/us it's a very very small chance. Maybe 1 in 500 thousand or something like that. I don't think the exact number really matters. –  Radu Murzea Dec 3 '13 at 7:31
    
you want to have a flush board at the same time? or just no clubs dealt to any person of the 5? thanks i will improve my answer when you tell me what you need. –  RayofCommand Dec 3 '13 at 11:33
    
@RayofCommand: I believe the question is, how often will the 5 club flush "board play" resulting in a chop, because no one has a club higher than the 2? –  Tom Au Dec 6 '13 at 16:53
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3 Answers

I think the real question here is with a 5 card flush on the board what is the chance that out of 5 players none have one of the remaining 8 flush cards? The answer is 12.3%

The odds of player one’s first card is not a flush suit is 39 of the remaining 47. We can see 5 cards, so 52-5 = 47 available cards. Of those 47 there are eight flush cards remaining so 47 – 8 = 39. That the second also is not a flush card would be 38 out 46. Multiply both fractions for the one player result of 0.685476. 68.5%.

Continue with player #2: 37/45 * 36/44 then multiply player one’s odds for a total of 46.1% for two players.

The third player has 35/43 * 34/42 then multiply the previous values for a total of 30.4%. The fourth player has 33/41 * 32/40 then multiply the previous values for a total of 19.6%. The fifth player has 31/39 * 30/38 then multiply the previous values for a final total of 12.3%.

To continue, if 10 players were still in, it is only a 0.7% chance of no other flushes.

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TL;DR Given a flush on the board, five players will split 19.7% of the time, or about once every 5 hands. Given a fresh deal, the deal will come up with a flush showing and a five-way split 0.0389% of the time, or once every 2,569 hands.


Some general probabilities

Overall odds that none of the five players have a given suit (say, hearts):

comb(39,10) / comb(52,10)
= 39/52 * 38/51 * 37/50 ... 30/43 (equivalent way to do this)
= 0.040186 (about 1 in 25 deals)

Overall odds that none of the five players have a given suit (say, hearts) and the board also has a flush of that suit (i.e., when you don't know whether the board will turn up with a flush):

(comb(13,5) / comb(52,5)) * (comb(39,10) / comb(52,10))
= 5/13 * 4/12 ... 1/9 * 39/52 * 38/51 * 37/50 ... 30/43
= 0.0000199 (about one in 50,251 total hands)

Overall odds that the board has a flush in any suit, but none of the five players have a card from that suit:

4 * (comb(13,5) / comb(52,5)) * (comb(39,10) / comb(52,10))
= 4 * 5/13 * 4/12 ... 1/9 * 39/52 * 38/51 * 37/50 ... 30/43
= 4 * 0.0000199
= 0.0000796 (about one in 12,562 total hands)

Some background math on how to calculate the odds of various flush-showing scenarios

Odds that none of the five players have a given suit (say, hearts) when you already know that the board also has a flush of that suit:

(comb(39,10) / comb(47,10))
= 39/47 * 38/46 * 37/45 ... 30/48
= 0.122777 (about one in 8 hands when there is a flush on the board)

This seems like it should be our answer, but we have to do some more math in second major section below. For example, it's possible to have a flush on the board and have one or more players have a card in that suit, but still chop because their card(s) are too low to play.


Okay, so what are the odds of an actual split when you know that the board is flush?

Figuring out the odds of a split pot given a flush on the board is much trickier. For example, if the board is A♥K♥Q♥J♥9♥, then the question is: what are the odds none of the players has the T♥? Obviously, any player with T♥ would win with a straight flush. Likewise, if the board is A♥K♥Q♥J♥5♥, then there's still a chop even if someone has the 4♥, but not if they have the 6♥.

Here's how we calculate this:

If the board has a 10-low flush (which would be a straight flush), odds of a split = 1.00000
If the board has a 9-low flush, odds of a split = comb(46,10)/comb(47,10) (46 = number of cards that are not showing and not the 10 of that suit) = .78723
If the board has a 8-low flush, odds of a split = comb(45,10)/comb(47,10) (45 = number of cards that are not showing and not higher than the 8) = .61609
If the board has a 7-low flush, odds of a split = comb(44,10)/comb(47,10) (44 = number of cards that are not showing and not higher than the 7 of that suit) = 0.47919
If the board has a 6-low flush, odds of a split = comb(43,10)/comb(47,10) (43 = number of cards that are not showing and not higher than the 6 of that suit) = 0.37028
If the board has a 5-low flush, odds of a split = comb(42,10)/comb(47,10) (42 = number of cards that are not showing and not higher than the 5 of that suit) = 0.28417
If the board has a 4-low flush, odds of a split = comb(41,10)/comb(47,10) (41 = number of cards that are not showing and not higher than the 4 of that suit) = 0.21651
If the board has a 3-low flush, odds of a split = comb(40,10)/comb(47,10) (40 = number of cards that are not showing and not higher than the 3 of that suit) = 0.16370
If the board has a 2-low flush, odds of a split = comb(39,10)/comb(47,10) (39 = number of cards that are not in that suit) = 0.12278

We need to know the odds of each of these outcomes. For example, if we know the board has a flush, what are the odds it's a 10-low (straight) flush? A 9-low flush? etc.

Odds 10-low = comb(4,4)/comb(13,5) = 0.000777000777000777
Odds 9-low = comb(5,4)/comb(13,5) = 0.00388500388500389
Odds 8-low = comb(6,4)/comb(13,5) = 0.0116550116550117
Odds 7-low = comb(7,4)/comb(13,5) = 0.0271950271950272
Odds 6-low = comb(8,4)/comb(13,5) = 0.0543900543900544
Odds 5-low = comb(9,4)/comb(13,5) = 0.0979020979020979
Odds 4-low = comb(10,4)/comb(13,5) = 0.163170163170163
Odds 3-low = comb(11,4)/comb(13,5) = 0.256410256410256
Odds 2-low = comb(12,4)/comb(13,5) = 0.384615384615385

So, the odds of a split given a flush on the board are:

P(10-low) * P(split|10-low) + P(9-low) * P(split|9-low) ... P(2-low) * P(split|2-low)
= comb(4,4)/comb(13,5) * comb(46,10)/comb(47,10) + comb(5,4)/comb(13,5) * comb(45,10)/comb(47,10) + ... + comb(12,4)/comb(13,5) * comb(39,10)/comb(47,10)
= 0.19653

So, the answer is that the probability of a 5-way chop when there is a flush on the board is .19653, or about 1 every 5 hands.

Note: technically, I have oversimplified very slightly here. It's possible for a player to have an in-suit card higher than the lowest card on the board and still get a chop, because the board could be a straight flush. For example, a player holds A♥ and the board is T♥9♥8♥7♥6♥ -- it's still a chop. The math here gets to be a pain, however, and has a very, very tiny impact on the overall calculation, so I've skipped it for simplicity. Some back of the envelope calculations puts the change at around 0.17% (P=0.001750364), or an additional 1 hand per 571, making the total probability 0.19828, or still around 1 in 5 hands.


Overall odds of a flush board and a five-way chop

We're almost to a general answer of the question, "How likely is it that a hand will have a flush board and split?" The answer is:

Number of suits * P(flush board|suit) * P(split|flush board)
= 4 * comb(13,5)/comb(52,5) * 0.19653
= 0.000389

In other words, starting from a fresh deal, approximately one hand in every 2,569 will be a five-way chop with a flush on the board.

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5 people holding 2 cards which are not from the same color. In other words, there are 13 cards which are not dealt to anyone out of 52 cards.

Lets make it easy - the probability that one person gets NOT a club in the first card given is 38/52 ~ 0.730 - the probability that one person gets NOT a club in the second card given is 37/51 ~ 0.725 ....

to calculate the chances you need to multiply all the chances after each other like this :

38/52 * 37/51 * 36/50 * 35/49 * 34/48 * 33/47 * 32/46 * 31/45 * 30/44 * 29/43 = 0.02988

remember the full probability is 1, meaning if the answer is 1 you get the card.

chance to get any card out of all 52 is 52 of 52 in other words 52/52 = 1

now you can divide 1 by 0.029 and see how often cards should be dealt to have average of 1 that this scenario happens ~ 34

so every 34 times this happens in average

if you want to have a flush board at the same time you need to calculate the situation as explained above. you have 42 cards left. (10 are dealt to players)

for a club flush board you need to calculate 14 / 42 * 13/41 * 12/40 * 11/39 * 10/38

since we don't know how many hearts, spades or diamonds are dealt to the people, it's not possible to calculate a flush board for that.

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Which makes the final odds of having a board with all clubs and none of the other five having a club 1 in 134,548 (calculated by multiplying the two numbers together). Note however, that the answer would be more often if you allowed the same scenario with any of the suits rather than just clubs (roughly four times as often). –  lnafziger Dec 3 '13 at 23:00
    
This is just wrong. The first part is incorrect: it should be 39/52 * 38/51, etc., not 38/52 * 37/51, etc. Likewise, 14/42 is incorrect: it's 13, and so on. Finally, you don't need to know how many hearts, spades or diamonds are dealt to the players to calculate the overall odds of a club flush with a split; that's irrelevant. –  Ed Cottrell Feb 12 at 18:05
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