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Page 1: Poker maths

Poker MathsBy Jim Makos

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Contents

• Using mathematics in poker• Expected Value• Pot Odds• Implied Odds• Calculating Expecting Value• Pot Equity• Fold Equity• Reverse Implied Odds

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Using mathematics in poker• A sound knowledge of

odds can only improve your game.

• Maths are most commonly used when a player is on a draw such as a flush or straight draw.

• Making the right decisions based on odds, you will be making more money in the long run.

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Expected Value

• The long-term average outcome of a given scenario.• Always make the decision that has the highest expected

value.• Maximizing +EV situations is often the difference

between a long term winner and a long term loser.

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Pot Odds

• The ratio of the amount of money actually in the pot compared to the amount of money required to call and maintain eligibility to win the pot, expressed with the pot amount first and calling amount second. 

e.g. 3:1 or 5:1

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Pot Odds Example

the_blues57 and 147_star’s pot odds are 1.33:1 given they need to call 2.5m to fight for the 3.3m pot. If the_blues57 calls, then 147_star’s pot odds will be 2.33:1.

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Implied Odds

The ratio of the amount of money that is expected to be in the pot by the end of the round or the end of the hand compared to the amount of money required to call and maintain eligibility to win the pot, expressed with the expected pot amount first and calling amount second. Different from pot odds because implied odds account for possible additional wagers.

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Implied Odds

Example

If the_blues57 calls, 147_star implied odds are better than 2.33:1 (pot odds), since they will likely win 15m chips more in case the turn is a nine.

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Calculating Expected Value

1. List all the possible outcomes of that action.

2. Find the probability and the win/loss of each outcome.

3. Put it all together in an equation and work it out.

Expected value∑ (probability to win)*win + ∑(probability to

lose)*loss

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Expected Value

Example

2 Scenarios:

1) the_blues57 calls, 147_star calls.

2) the_blues57 calls, 147_star calls assuming villains won’t bet the turn.

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Scenario 1:

Given we know the_blues57 holds a nine, the probability for 147_star to hit his straight on turn is about 14%. If he does, he will win at least 8.3m. Thus,

Expected value = (probability to hit the straight) * win/loss + (probability to miss the straight) * win/loss

EV = 14% * 8.3m + 86% * (-2.5m) = -1m

Scenario 2:

Assuming 147_star will see the river card without putting any more chips on turn, the probability to hit his straight is doubled.

EV = 28% * 8.3m + 72% * (-2.5m) = +0.5m

Scenarios1) the_blues57 calls, 147_star calls.

2) the_blues57 calls, 147_star calls assuming villains won’t bet the turn.

Conclusion147_star should only call if he strongly believes that the opponents will play passively on turn.

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Pot Equity

• The average amount of money that a particular hand would win if the specific situation were repeated a large number of times; at a given point, the amount of money at stake in the pot multiplied by the percentage chance of winning. Different from expected value because it does not account for the cost of additional wagers.

e.g. 15% or 38%

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Pot Equity Example

Given that the hole cards would be unknown to 147_star, he would assume that their chance of winning the pot is 32%, or his pot equity is about 1.9m chips on flop. If hole cards were revealed, Benjamin89 would need to hit an Ace or a Queen to win the hand, without the_blues57 hitting another eight or a flush and 147_star missing his straight draw. Benjamin’s win probability on flop is about 17%, blues’ 50% and star’s 32%. Thus, the pot equity is:

Benjamin: 1m, Blues: 2.9m, Star: 1.9m

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Fold Equity• Fold equity is the additional equity you

gain in the hand when you believe that there is a chance that your opponent will fold to your bet.

• If we think it is likely that our opponent will fold to our bet, we have a lot of fold equity.

• If we think it is unlikely that our opponent will fold to our bet, we have little fold equity.

• If we do not think our opponent will fold to our bet, we have no fold equity.

How much fold equity do we have?

(chance our opponent will fold)*(opponent's equity in the hand)

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Fold Equity Example

Suppose the_blues57 calls and 147_star believes there is 50% chance that his opponents will fold if he shoves.

Fold Equity = 50% * (17+50) = 33.5%

Total Equity = Pot Equity + Fold Equity = 32+33.5 = 65.5%

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Reverse Implied Odds

Reverse implied odds are reduced pot odds that include future losses that could occur if an opponent has or gets the upper hand.

Reverse implied odds are the opposite of implied odds. With implied odds you estimate how much you expect to win after making a draw, but with reverse implied odds you estimate how much you expect to lose if you complete your draw but your opponent still holds a better hand.

Reverse implied odds are how much you could expect to lose after hitting your draw.

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Reverse Implied Odds

Example

A nine completes the straight for 147_star. However if someone’s hand was JT, then he would certainly lose all his stack to the best hand, in case turn or river was a nine.

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Bibliography and References

The theory of poker – David Sklansky Easy Game Vol. I & II – Andrew Seidman Hold’em Poker for Advanced Players –

David Sklansky & Mason Malmuth Two Plus Two Forum PokerStrategy.com Pictures provided from pokerstars.com