# Holdem probabilities

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I realize this may be slightly off topic but I was pondering a question regarding probabilities and Texas Holdem. For those who aren't aware of how Texas Holdem works it is played from a deck of 52 cards, each player holds 2 cards each and the table can hold up to 5 which means that for 2 players there will be 9 cards in play at the end. Now for my question regarding probabilities, let's say we have the following scenario:
1. We know the number of cards in our hand and on the table.
2. We know the odds of beating a single opponent if the game goes to the end (i.e 7 cards become known to us).
How can we from the knowledge of what our odds of beating 1 opponent calculate the odds of beating an arbitrary number of opponents? Some examples (based on 1,000,000 Monte Carlo simulations so may be off by 1 - 2%): 1 opponent, 5 cards dealt, odds of winning against 1 opponent is 0.38:
• 2 opponents: 0.24
• 3 opponents: 0.19
• 4 opponents: 0.17
• 5 opponents: 0.15
• 6 opponents: 0.14
• 7 opponents: 0.13
• 8 opponents: 0.12
• 9 opponents: 0.21
1 opponent, 6 cards dealt, odds of winning against 1 opponent is 0.84:
• 2 opponents: 0.72
• 3 opponents: 0.61
• 4 opponents: 0.52
• 5 opponents: 0.44
• 6 opponents: 0.38
• 7 opponents: 0.32
• 8 opponents: 0.27
• 9 opponents: 0.23
1 opponent, 2 cards dealt, odds of winning against 1 opponent is 0.59:
• 2 opponents: 0.42
• 3 opponents: 0.33
• 4 opponents: 0.28
• 5 opponents: 0.24
• 6 opponents: 0.2
• 7 opponents: 0.18
• 8 opponents: 0.16
• 9 opponents: 0.14
1 opponent, 5 cards dealt, odds of winning against 1 opponent is 0.83:
• 2 opponents: 0.7
• 3 opponents: 0.59
• 4 opponents: 0.5
• 5 opponents: 0.43
• 6 opponents: 0.36
• 7 opponents: 0.31
• 8 opponents: 0.27
• 9 opponents: 0.23
5 cards dealt means that each player has 2, table has 3. Probabilities are of what the chance is to have the winning hand once 2 more cards have been dealt. Initially I thought this would be fairly trivial to solve but after a few hours I'm starting to suspect either my Monte Carlo simulation is wrong or it's not trivial for me :) The actual trends within each series is pretty easy to get but devising an equation which holds independently of number of opponents, initial odds and number of dealt cards seems not to be.

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I don't believe there is a simple equation. Some hands are much better against many players (e.g., suited connectors) and much hands are much better against few players (e.g., ace king unsuited).

What's wrong with using Monte Carlo to compute these probabilities? That's how I would do it...

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Rutin
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