This document discusses various poker math concepts including expected value, pot odds, implied odds, calculating expected value, pot equity, fold equity, and reverse implied odds. It provides examples and explanations of each concept to help players incorporate mathematics into their poker strategy and make optimal decisions based on odds. Understanding these concepts can improve a player's game and increase their profits over the long run.

1. Poker Maths
By Jim Makos

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

3. 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.

4. 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.

5. 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

6. 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.

7. 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.

8. 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.

9. 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

10. 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.

11. Scenarios
1) the_blues57 calls, 147_star calls.
2) the_blues57 calls, 147_star calls assuming villains won’t bet the turn.
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
Conclusion
147_star should only call if he strongly believes that the
opponents will play passively on turn.

12. 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%

13. 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

14. 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.
(chance our opponent will fold)*(opponent's equity in the hand)
How much fold equity do we have?
• 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.

15. 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%

16. 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.

17. 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.

18. 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