The Rule of 2 and 4 are quick shortcuts to help us work out the percentage needed to get a draw in Texas No-Limit Hold'em.  These shortcuts aren’t exact.  However, they are good enough for doing quick probabilities in our head at the tables.

The Rule of 2
The Rule of 2 states: Multiply your number of outs by 2 to get the approximate percentage of a draw you have on the turn or the river.   

For example, let's say you have a flush draw after the flop. 

• There are 2 cards of the same suit in your hand and 2 cards of the same suit on the board. 
  That makes 4 cards to a flush. 
• There are 13 cards to a suit, and 9 cards (outs) that can help you make a flush (13 - 4 = 9).
• Use the Rule 2 to approximate your percentage of hitting a flush on the next card.
  2 x 9 = 18% probability of getting your flush. 
  
  

The exact calculation is:

• There are 13 cards to a suit, and 9 cards (outs) that can help you make a flush (13 - 4 = 9).
• There are 52 cards in the deck minus 2 cards in your hand and 3 cards on the board.
• This leaves 47 cards remaining (52 - 5 = 47).
• (9/47) = 19% probability of getting your flush.

The Rule of 2 works because there are 52 cards in the deck
 and 52 goes into one hundred approximately twice.

The Rule of 4
The Rule of 4 is used when you are considering an all-in move after the flop and will see both the turn and river cards.  Multiply your number of outs by 4 to get the approximate percentage of a draw.
 
Like the Rule of 2 example,  you have a flush draw after the flop. 

• There are 2 cards of the same suit in your hand and 2 cards of the same suit on the board. 
  That makes 4 cards to a flush. 
  There are 13 cards to a suit, you have 9 cards (outs) remaining that can help you make your flush (13 - 4 = 9).
• Use the Rule 4 to approximate your percentage of hitting a flush on the next two cards.
  4 x 9 = 36% probability of making your flush. 
  
  

The exact calculation is as follows:

• There are 52 cards in the deck minus 2 cards in your hand and 3 cards on the board. 
  Consequently, there are 47 cards remaining (52 – 2 – 3 = 47).
• There are 9 cards that will help you make your flush. 
  Thus, there are 47 - 9 = 38 cards that will not make your flush on the turn.
• The probability of not making your flush on the turn is 38/47.   
• Subtract one from the numerator and denominator for the river card which is 37/46.
• Multiply both numbers together and subtract from 1 to determine your probability of making a flush.
• 1- 38/47 x 37/46 = 35% probability of making your flush.

Once again, that’s close and useful for quick estimate at the tables.

Note: The Core Math Chapter of Mastering Poker Math provides more detail on how this equation is derived.

Pot odds and implied odds are major concepts in the math of poker.  It is crucial to know when the odds are in your favor, and when they are against you.  Then, you can make informed decisions on when to call, bet, fold, raise, re-raise or go all-in. 
Note: For the examples in this section, round numbers will used (whenever possible).  It is easier to learn a concept in black and white.  The real world runs in shades of gray.  Consequently, at the poker tables you will need to approximate.  

Pot Odds
Pot odds are the ratio of the current size of the pot to the cost of a contemplated call. 
For example, if there is $7,500 in a pot and villain (an opponent) bets $2,500.  There is now $10,000 in the pot.  It will cost you $2,500 to win a potential $10,000.  If you bet there will be $12,500 in the pot. 

Your pot odds are 20% ($2,500/$12,500).  You must win 1 out of 5 pots to break even.  This can be expressed as 1:4.   That is to say, you must win one pot and can lose four pots to break even.  If you win more than that, you will make money.   

When the odds of drawing a card that wins the pot are higher than the pot odds, the call has a positive expectation.  In other words, on average, you will win more money than the cost of the bet.

Conversely, if the odds of drawing a winning card are numerically lower than the pot odds, the call has a negative expectation.  On average, you will win less money than the cost of the bet.

To explain this concept further let’s review a flush draw.  Sometimes this is called chasing a flush.

Example #1: Getting Pot Odds on a Flush Draw

• You have Ace 8 suited (hearts). 
• The flop has 3 different cards with two hearts.
• After the flop villain bets $1,500 into a $8,500 pot.  The pot is now $10,000.
• You must put in $1,500 to call the bet (making the pot $11,500).
• You have a 19% probability to make a flush on the next card.
• You have pot odds of 13% ($1,500/$11,500).  This gives you an  expectation of  +6% (19% - 13%) over the long run. Also, you have good implied odds which will be covered next. 
• This is a good bet mathematically (assuming you aren’t short stacked and/or have other considerations that will encourage you to fold).   

Implied Odds
Implied odds are your potential winnings by the end of the hand compared with the amount of money required to make a call.  They are different than pot odds because they account for possible future betting.

Implied odds are calculated in situations where you have a draw.  You expect to make money on additional bets if your draw is made.  And, you can fold to a bet on the next card if the pot and implied odds are unfavorable.

Since you expect to gain additional money in a later round, or rounds (if your draw is made, and not be committed to lose additional bets when your draw is missed), the extra money you expect to gain is added to the current pot size. 

What you may gain is only an estimate depending on your read of the other player.  It isn’t an exact science. 

Example #2: Getting Implied Odds on a Flush Draw

• You again have Ace 8 suited (hearts).  
• The flop has 3 different cards with two hearts.
• Villain bets $3,500 into a $6,500 pot.  The pot is now $10,000.

• You must put in $3,500 to call the bet (making the pot $13,500).
• You have a 19% probability to make a flush on the next card.
• You have pot odds of 26% ($3,500/$13,500).   This gives you an expectation of  -7% (19% - 26%) over the long run.
• However, you consider making the call since you have good Implied odds.
  • Implied odds are important if you think you can get more money from your opponent(s) if you hit your flush on the following round/rounds of betting (assuming you aren’t short stacked or have other considerations).  This is especially true in this example because you have the potential for a nut flush.
    
    

 Example #3: Not Getting Pot Odds on a Flush Draw

• You again have an Ace 8 suited (hearts).  
• The flop has 3 different cards with two hearts.
• Villain bets $6,000 into a $4,000 pot.  The pot is now $10,000.
• You must put in $6,000 to call the bet (making the pot $16,000). 
• You have a 19% probability to make a flush on the next card.

• You have pot odds of: 38% ($6,000/$16,000).    You are getting a potential 19% for a cost of 38%.  This gives you an expectation of -19% (19% - 38%) over the long run.

• Mathematically, this is a bad deal and you should consider folding.

To learn more please go to the: Mastering Poker Math Book

Copyright 2021
Charles W. Clayton
All Rights Reserved
 

The variability in poker is huge.  This gives rise to some fascinating characteristics of the game. 

First, the high variance of the game provides excitement that has some addictive qualities.

Second, a high percentage of players think they are better than they are because they can play with excellent players and do okay at times.  They play by their gut and rarely (if ever) use the math.  The high variability allows them this luxury.

And third, there are few (possibly no other) major sports or games where the variability is as high as it is in Texas No-Limit Hold’em.  Here are a couple of examples.
 

• If you worked hard on your golf game for three years solid what are your chances of winning against Tiger Woods, even on his worst day?  It is essentially 0%.

 

• If you worked diligently on competitive swimming for a few years, what is your probability of beating Michael Phelps in the pool on his worst day?  Also, it is essentially 0%.

 
The point is that in most sports and games, the top players have an overwhelming advantage (because of small variances) over beginners in the long run, and well as in the short-run.  This isn’t the case for No-Limit Hold’em in the short-run.
 
Tight Hand Variances
The poker hands in Hold’em have relatively tight hand variances.   Here are some examples:
 

• AKs (a premium hand) is a 69% to 31% favorite over 7,2o (the worst hand in poker) before the flop.  In other words, AKs will only win 2 out of 3 times if the hand goes to showdown.

 

• KQs (a premium hand) against 6,5s (a highly speculative hand) is only a 61% to 39% favorite before the flop.  Thus, KQs will only win about 3 out of 5 times if the hand goes to showdown.

 

• 6,5s against AA (the best starting hand) has a 23% probability of winning before the flop.  Consequently, 6,5s will win almost 1 out of 4 times if the hand goes to showdown. 

 
Note:  The “s” in these examples means suited.  The “o” means non-suited.  This convention will be used throughout the text for suited and unsuited cards.

There are many more examples.   The point is, even the worst hands will win a reasonable percentage of the time against the best hands.

It is this tight variance in poker hands which allows bad players to get lucky in the short run and beat good players.  This is a major reason why vast numbers of people enjoy playing the game.  On any given day even some of the worst players have a slight  chance to win.

But, not in the long run!  The best players will consistently win because the game of Texas No-Limit Hold’em is really a skill game, not a luck game.

Self-delusion is common in poker because of occasional runs of good cards and good luck. But good cards dry up, and good luck runs out. 

Bad beats (low probability events) by bad players are common in poker and cause many tilt related rantings, especially from those who don’t understand the math.  Yet, it is just the variance which provide these emotional roller-coaster rides all poker players have experienced time and again.
 
Millions of people play the game because of the high variability   allowing them to get lucky at times.  They supply massive sums of money for the tournaments fueling the poker boom.  The silver lining is that the exceptional players do well over the long run.