Poker maths

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A presentation regarding the use of mathematics in poker.

Transcript of Poker maths

  • 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 ofodds can only improveyour game. Maths are most commonlyused when a player is on adraw such as a flush orstraight draw. Making the right decisionsbased on odds, you will bemaking more money in thelong run. 4. Expected Value The long-term average outcome of a given scenario. Always make the decision that has the highest expectedvalue. Maximizing +EV situations is often the differencebetween a long term winner and a long term loser. 5. Pot Odds The ratio of the amount of money actually in the potcompared to the amount of money required to call andmaintain eligibility to win the pot, expressed with thepot amount first and calling amount second.e.g. 3:1 or 5:1 6. Pot OddsExamplethe_blues57 and 147_stars pot odds are 1.33:1 given they need tocall 2.5m to fight for the 3.3m pot. If the_blues57 calls, then147_stars pot odds will be 2.33:1. 7. Implied OddsThe ratio of the amount of money that is expected to bein the pot by the end of the round or the end of the handcompared to the amount of money required to call andmaintain eligibility to win the pot, expressed with theexpected pot amount first and calling amount second.Different from pot odds because implied odds accountfor possible additional wagers. 8. Implied OddsExampleIf the_blues57 calls, 147_star implied odds arebetter than 2.33:1 (pot odds), since they willlikely win 15m chips more in case the turn is anine. 9. Calculating ExpectedValue1. 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. ExpectedValueExample2 Scenarios:1) the_blues57 calls, 147_star calls.2) the_blues57 calls, 147_star callsassuming villains wont bet the turn. 11. Scenarios1) the_blues57 calls, 147_star calls.2) the_blues57 calls, 147_star calls assuming villains wont bet the turn.Scenario 1:Given we know the_blues57 holds a nine, the probability for 147_star to hit hisstraight 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 missthe straight) * win/lossEV = 14% * 8.3m + 86% * (-2.5m) = -1mScenario 2:Assuming 147_star will see the river card without putting any more chips onturn, the probability to hit his straight is doubled.EV = 28% * 8.3m + 72% * (-2.5m) = +0.5m Conclusion147_star should only call if he strongly believes that theopponents will play passively on turn. 12. Pot Equity The average amount of money that a particular hand would win ifthe specific situation were repeated a large number of times; at agiven point, the amount of money at stake in the pot multiplied bythe percentage chance of winning. Different from expected valuebecause 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. Benjamins win probability on flop is about 17%, blues 50% and stars 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)*(opponents 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 offold equity. If we think it is unlikely that our opponent will fold to our bet, we have littlefold 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 OddsReverse implied odds are reduced pot odds that include future losses thatcould occur if an opponent has or gets the upper hand.Reverse implied odds are the opposite of implied odds. With implied odds youestimate how much you expect to win after making a draw, but with reverseimplied odds you estimate how much you expect to lose if you complete yourdraw but your opponent still holds a better hand.Reverse implied odds are how much you could expect to lose after hittingyour draw. 17. ReverseImplied OddsExample A nine completes the straight for 147_star. However if someones 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 Holdem Poker for Advanced Players DavidSklansky & Mason Malmuth Two Plus Two Forum Pictures provided from