|
Sun137 posted this in the poker forum and it's just too valuable not to share here as well. These are some basic math concepts for players who have not yet studied this part of their poker game and strategy. In fact, you may think you know poker math but in spite of that, you could still use this as a reference when needed.
THE STUDY OF BASIC FUNDAMENTALS
1. The definition of Good Poker
GOOD POKER IS:
The ability to accurately quantify, or guess, the likelihood of your opponents actions and use mathematics to determine the line that will have the highest possible average rate of return.
PART 1
Anticipate your opponent’s actions with good assumptions (improve with experience)
PART 2
Finding the best line (use mathematics)
= Making the most money in poker
Traditional route is via forums, asking questions and receiving views - possibly conflicting or wrong.
2. Ratios/Fractions/Percentages
A RATIO describes one thing in comparison to another
To convert a ratio to a fraction, add both numbers together to find the whole - eg in 2:1 there are 3 total chances. The FIRST event will happen 2/3 of the time and the SECOND event will happen 1/3.
A FRACTION describes something as part of a whole
To convert a fraction to a ratio SUBTRACT the first number frrom the second number to find what amount IS NOT the first event - eg in 1/3 there are 2 times it does NOT happen. This figure is the ratio, in this case 2:1
A PERCENTAGE describes the same thing as a fraction but the whole is 1.0 or 100%
To convert a fraction to a %, divide the first number by the second number and multiply by 100 - eg 1/3 = 0.33 = 33%
To convert a % to a fraction, divide 100 by the % to find out of what whole you will have 1 event - eg 33% = 33/100 = 1/3
Multiply top (numerator) and bottom (denominator) numbers in fractions
Simplify by dividing top AND bottom by the top number.
3. The % chance for 1 card to come
To find the % chance for 1 card to come, divide the number of cards of that type by the number of unknown cards (pre-flop = 50, flop = 47, turn = 46) – for example, the chance of an Ace on the turn if none has appeared in your hole cards or on the flop is 4/47.
The formula for multiple chances (turn and river) is:
1 - (c1/u1)*(c2/u2) where c = chances of failure and u = unknown values. We subtract the chances of the cards NOT being of the type required from 1.
Using the Ace example above, the formula would be:
1 - (43/47)*(42/46)
4. Calculating Combinations
A combination describes the possible arrangement of the parts of a group within a given set of parameters. A poker hand is 1 combination (combo) within a group (Sequences do NOT matter).
HAND COMBOS (NOT USING BOARD CARDS)
There are 2 types of hand - non pocket pairs and pocket pairs.
NON POCKET PAIR HANDS
Combos = Number of unknown cards of a type x number of unknown cards the first card can be combined with in order to create a hand that is within the group - eg AK has 4 unknown Aces and 4 unknown Kings = 4 x 4 = 16 possible combos.
This is made up of 4 SUITED combos and 12 UNSUITED combos
ANY non PP Hand = 16 x the number of possible kickers
Any 2 BW (AKQJT) = 10 combos (4+3+2+1)
POCKET PAIR HANDS
4 unknown cards = 6 combos
3 unknown cards = 3 combos
2 unknown cards = 1 combos
HAND COMBOS (USING BOARD CARDS)
1 Pair = 3x4 = 12 combos x the number of possible kickers (using 1 board card)
2 Pair = 3x3 = 9 combos (2 board cards)
2 Pair = 27 (3 board cards)
2 Pair = 54 (4 board cards)
2 Pair = 90 (5 board cards)
A set = 3 combos (1 board card)
OESD = 16 combos (2 board cards)
Flush draw = 55 combos (2 board cards)
Flush draw = 45 combos (3 board cards)
Any 2 from 11 = 55 (10+9+8+7+6+5+4+3+2+1)
Any 2 from 10 = 45 (9+8+7+6+5+4+3+2+1)
Any 2 from 9/8/7/6/5/4/3 = 36/28/21/15/10/6/3 combos
In any poker problem, identify all the possible combos and place them in Groups. Add all the Group combos together to obtain a TOTAL combo figure. The chance of an opponent holding 1 group of hand combinations (combos) is equal to Group Combos/Total Combos
WEIGHT combos if the chances of occurring are different:
Weighting = Number of Combos x % chance of them occurring
This will adjust the figure for TOTAL number of combos used and the results for the % chance of any one group occurring.
5. The calculation of EV
Expected Value (EV) describes the average amount of chips/($) won or lost by that strategy (any actions or series of actions).
The normal goal is to always choose the strategy with the highest EV.
EV requires 3 elements:
1. How often you win the money in the pot*
2. How often you win your opponent(s) bet(s)
3. How often you lose your bet(s)
* This is YOUR EQUITY (note: Hot/Cold equity only refers to ‘everyone all-in’ scenarios)
The standard formula is:
EV = ($ won from pot) + ($ won from bets) - ($ lost from bets)
Example:
A $10 bet made by you
$20 pot
Opponent folds 40%
Opponent calls 50% - you win 70% and lose 30%
Opponent raises $50 x 10% - you win 35% and lose 65%
Pot win% = 0.40 + (0.50*0.70) + (0.10*0.35) = 0.785 or 78.5%
Therefore Step 1 ($ won from pot) is $15.70 ($20*0.785) for a $10 bet.
Step 2 ($won from bets)
Identify how often, on average, you WIN the pot when the opponent(s) put various amounts of money into the pot and then find the average:
Opponent Calls = $10 x 70% win = $7 x 50% of the time = $3.50
Opponent Raises = $50 x 35% win = $17.50 x 10% of the time = $1.75
Therefore Step 2 ($ won from bets) is $5.25 [($7*0.50) + $17.5*0.10)]
Step 3 ($lost from bets)
Identify how often, on average, you LOSE the pot when YOU put various amounts of money into the pot and then find the average.
Opponent Calls = $10x30% lose = $3 x 50% of the time = $1.50
Opponent Raises = $50x65% lose = $32.50 x 10% of the time = $3.25
Therefore Step 3 ($ lost from bets) is $4.75 [($3*0.50) + ($32.50*0.10)]
Add the results together:
EV = $15.70 + $5.25 - $4.75 = $16.20
If the EV strategy comparison does not involve anyone folding, ignore Step 1 ($won from pot) and the formula becomes:
EV = $won from bets - $lost from bets
(Note: this is because the average amount of money won from the pot will be the same for either strategy)
A shortcut for Step 3 ($lost from bets) is to subtract the $won from bets value from the total value of the bet you are going to make:
$10 - $5.25 = $4.75
6. Calculating All-in EV
How to calculate the EV of an All-in Bet or Raise
There are 4 steps:
1. Estimate what range of hands your opponent could hold
2. Estimate what hands your opponent calls an all-in raise with and determine what portion of his or her total range these hands represent (using combos)
3. Find your average equity against your opponent's calling range
4. Find how much money is in the pot before your raise, the cost of your raise and the cost for your opponent to call your raise
Example 1 - One Opponent
Blinds $15
You Open raise $35
Opponent re-raises $160
You plan to raise all-in with $1000
Step 1 – You estimate the opponent will only raise with 15% of hands
Step 2 – You estimate the opponent will only call all-in with 88+/AQ+. This is 74 hands (42 pairs/32 non pair) = 5.6% of all possible holdings. This represents 37% (5.6/15) of his re-raising range.
(Note: 1 hand = 1/1326 = 0.075% = 3/40. A rough conversion from % to hands is to multiply by 13, from hands to % is to multiply by 3/4 and divide the answer by 10 – or just divide by 13).
Step 3 – Average Equity v the Opponent Calling Range is 31%.
(Note: This is found by using Pokerstove but can be calculated manually)
Step 4 – The Pot before all-in is $210 ($160 + $35 + $15) , the all-in raise cost is $965 ($1000 - $35) and the opponent call cost is $840 ($1000 - $160).
Use the standard EV formula from Section 5 above:
EV = ($won from pot) + ($won from bets) – ($lost from bets)
Calculate how often the pot will be won:
Opponent calls 37% * 31% equity = 12%, added to 63% opponent folds = 75% of the time the pot will be won.
Step 1: $won from pot = 75% * $210 = $157.50
Step 2: $won from bets = ($840 opponent call cost) * (37% opponent calls) * (31% equity) = $96.35
Step 3: $lost from bets = ($965 your bet) * (37% opponent calls) * 69% NO equity) = $246.35
Total EV = $157.50 + $96.35 - $246.35 = $7.50
Example 2 - MULTIPLE OPPONENTS
Calculate each scenario separately before adding the results together to obtain an overall average.
Blinds $3 (including $2 from BB Opponent 1)
Pre-flop $24 (your $8 raise and 2 calls)
Flop $70 (Opponent 1 bets $20, Opponent 2 raises $50)
Pot = $95 ($70 bet on Flop + $24 bet Pre-Flop + $1 SB)
Call = $50
All-in = $192 v Opponent 1 and $122 v Opponent 2
Opponent 1 Stack = $200 ($172 left)
Opponent 2 Stack = $130 ($72 left)
Your Stack = $200 ($192 left)
Opponent 1 will fold 75% of time to an all-in
Opponent 2 will fold 30% of the time if Opponent 1 folds and 40% of the time if Opponent 1 calls
Your Equity calculations for the 4 options (using Pokerstove):
35% v Opponent 1 calls and Opponent 2 folds (10%)
45% v Opponent 1 folds and Opponent 2 calls (52.5%)
31% v Opponent 1 AND Opponent 2 call (15%)
100% v Opponent 1 AND Opponent 2 FOLD (22.5%)
The %ages in brackets are the combined opp1/opp2 call/fold actions:
Call/Fold = 25% x 40% = 10%
Fold/Call = 75% x 70% = 52.5%
Call/Call = 25% x 60% = 15%
Fold/Fold = 75% x 30% = 22.5%
Calculate how often the pot will be won (calculate each of the 4 options separately):
10% Opp1 only calls x 35% your Equity = 3.5%
52.5% Opp2 only calls x 45% your Equity = 23.625%
15% Opp1 & 2 calls x 31% your Equity = 4.65%
22.5% Opp1 & 2 fold x 100% your Equity = 22.5%
Total % pot won = 54.275%
Step 1: Total value of pot won = $95 x 54.275% = $51.56
Step 2: $ won from bets:
Opp1: $172*10%*35% = $6.02
Opp2: $72*52.5%*45% = $17.01
Both: $244*15%*31% = $11.35
Total won = $6.02 + $17.01 + $11.35 = $34.38
Step 3: $ lost from bets:
Opp1: $192*10%*65% = $12.48
Opp2: $122*52.5%*55% = $35.23
Both: $192*15%*69% = $19.87
Total lost = $12.48 + $35.23 + $19.87 = $67.58
TOTAL EV = $51.56 + $34.38 - $67.58 = $17.80
Always remember to weight the expected gain or loss by the % chance of the event happening.
If an EV is profitable, it may NOT be the MOST profitable as there may be a more profitable scenario (including folding).
EV may operate in isolation in a cash game, but in multi-table poker tournament there may be tournament considerations that override EV in the final decision on what action to take.
7. Pre Flop Bluffing
Calculate how often your opponent needs to fold in order for a bluff to show an immediate profit.
Immediate profit = When your opponent folds (it does not consider anything else (for example, your possible outs to improve).
This is useful for a pure bluff or bluffs where your opponent will either fold or raise, when you will fold.
No EV calculations are necessary, just determine, based on price, how often he needs to fold to make a bluff profitable.
The price = Money in pot : Cost of Bluff
This is a ratio and can be converted into a % by the formula:
Cost of bluff / (Money in pot + Cost of bluff)
Example: In this round
Blinds $5 (1 Bet)
Hero open raise = $15 (2 Bet)
Villain re-raise = $45 (3 Bet)
Pot is now = $65
A 4-bet bluff raise to $110 = $95 cost
Ratio = $65 : $95 = 95/(95+65) = 95/160 = 59%
Therefore the opponent needs to fold more than 59% of the time to make the bet IMMEDIATELY profitable.
Therefore the opponent has a MAXIMUM CALLING % of 41% to make the bluff profitable (100% - 59% folding range).
AND
You now determine how wide your opponent's pre-flop range needs to be in order for a pre-flop bluff to show an immediate profit.
This is achieved by the formula:
% of hands opponent will CALL / Maximum % opponent can call before the bluff becomes unprofitable.
(Estimate the hands opponent will call with and Use Pokerstove to determine what % of all possible hands this is - or use the Number of Hands / 13 shortcut).
Subtract the MINIMUM % you need your opponent to fold from 100% to find the MAXIMUM amount your opponent can call before the bluff becomes unprofitable (This has already been done above).
Example (continued from above)
You Estimate the Opponent will call with 88+/AJs/AQ+, or 5.9% of all possible hands (42 pairs, 4 suited, 32 non suited = 78 hands/1326 hands)
[A very close shortcut is to divide 78 by 13 = 6%]
% hands opponent will call = 5.9% (say 6%)
Maximum allowed % range for opponent calling = 41%
6/41 = 14.4%
Therefore the bluff will show immediate profit if the opponent holds 14.4%+ of all possible hands.
8. All-in Bluffing
Calculate how often your opponent needs to fold to make an all-in semi-bluff profitable.
The ALL-IN BLUFF IMMEDIATE PROFIT formula is:
RATIO: Money in pot : Cost of Bluff OR
PERCENTAGE: Cost of Bluff / (Money in Pot + Cost of Bluff)
The SEMI BLUFF requires the cost of bluff to be the 'Average Loss when called'
There are 3 steps:
1. Estimate the hands your opponent will call/raise an allin bet.
2. Use Pokerstove to determine the average equity
3. Calculate the EV when called
EXAMPLE
HU Game (One opponent)
Hero 3 bets to $10
Villain 4 bets to $25
Pot is now $35
Hero Stack $100 ($90 remaining)
Villain Stack $100 ($75 remaining)
Step 1: You estimate the opponent will call an all-in raise with TT+/AQ+ (30 pair hands + 32 non pair hands = 62 hands or 4.7%)
Step 2: Pokerstove indicates you have 37% equity
Step 3: EV is the standard formula:
($35*37%) + ($75*37%) - ($90*63%) = -$16
A loss of $16 - this is the Average Loss when Called, which replaces the Cost of Bluff in the normal All-in Ratio Formula.
RATIO = Money in Pot : Average Loss when Called is:
$35 : $16 = 16/(35+16) = 16/51 = 31%
Therefore the opponent needs to FOLD more than 31% of the time to make your all-in semi-bluff profitable.
Therefore the opponent has a MAXIMUM CALLING % of 69% to make the all-in semi-bluff profitable (100% - 31% folding range).
AND
You now determine how wide your opponent's pre-flop range needs to be in order for an all-in pre-flop semi-bluff to be profitable by the formula:
% of hands opponent will CALL / Maximum % opponent can call before the all-in semi bluff is unprofitable
(Estimate the hands opponent will call with and Use Pokerstove to determine what % of all possible hands this is - or use the Number of Hands / 13 shortcut).
Subtract the MINIMUM % you need your opponent to fold from 100% to find the MAXIMUM amount your opponent can call before the all-in semi-bluff becomes unprofitable (This has already been done above).
Example (continued from above)
% hands opponent calls = 4.7%
Max % opponent can call = 69%
4.7/69 = 6.8%
Therefore the all-in semi-bluff will be profitable if the opponent holds 6.8%+ of all possible hands
9. The Nash Equilibrium Strategy
The Nash Equilibrium Strategy is a Game Theory Optimal Strategy by John Forbes Nash.
It is a strategy which will win the most money possible against an opponent who always chooses the best possible counter-strategy to yours.
EXAMPLE
In a Rock/Paper/Scissors game, the NES is to make a random selection of 33% for each item against an opponent who ALWAYS chooses the best counter-strategy to yours. (He doesn't see what you pick, he just determines the best strategy based on your strategy).
The difference between exploitative strategies and equilibrium strategies
We expect virtually ALL players will NOT choose the optimal counter-strategy. Therefore the NES is NOT the optimum to use.
You need to play an EXPLOITATIVE Strategy which has the optimum possible expectation.
An EXPLOITATIVE Strategy refers to any strategy with a higher expectation than a NES against a particular opponent.
EXAMPLE
If an opponent chooses ROCK 45% of the time, you should always choose PAPER and expect to win 45% of games as opposed to 33% in the NES equal random selection.
(NOTE: ALL non-equilibrium strategies are exploitable (including yours) BUT 2 opponents cannot exploit each other AT THE SAME TIME)
EXAMPLE
You bet $50 into a $100 pot on the river
The pot is laying you 2:1 on your bluff
The pot is laying him 3:1 on his call
$100:$50 = 2:1 = 1/3 = 33% for your bluff therefore the opponent's NES is to call 67% of the time.
$150:$50 = 3:1 = 1/4 = 25% for his call, therefore your NES is to bluff 25% of the time
If the opponent calls less than 67% of the time (his NES), you can exploit him by bluffing more than 25% (your NES) - if he calls more than 67%, you can exploit him by bluffing less.
If you bluff less when he calls less, or bluff more when he calls more, HE will be exploiting YOU - even if he is not exploiting you OPTIMALLY.
You can use the pot : bet ratio to establish the NES and determine, based on your opponent's play, whether to increase the level of bluffing or decrease it.
You can also do the reverse - determine from his play, what is the best bet size to exploit your opponent.
10. Solving Problems
Every poker decision has 2 parts:
Make assumptions about how likely your opponent is to take particular actions. This is an INDUCTIVE process based on experience, discipline and rational decision making.
Choosing the strategy to provide the highest EV based on those assumptions. This is a DEDUCTIVE process, as for any set of assumptions you can prove mathematically what the best strategy is and with enough prowess you can perform this step perfectly the majority of the time.
To solve EXPLOITIVE poker problems you only need 2 major skills:
The ability to count and compare combos
The ability to perform EV calculations
GOOD ASSUMPTIONS + PERFECT LINES = $$$
|
Blog 
