However, most of the time we want to see this in hand odds, which will be explained after you read about pot odds. To change a percent to odds, the formula is: Odds = ( 1 / Percentage ) - 1 Thus, to change the 24% draw into an odd we can use, we do the following: Odds = ( 1 / 24% Two Overcard Draw ) - 1 = ( 1 / 0.24 ) - 1 = 4.17 - 1 = 3.17 or approx 3.2

ANALYZING PROBABILITIES IN DEPTH

You may want to skip this section and go back to it later. Some of this is pretty deep. But important! Getting a handle on the probability of being dealt various poker hands is one of the most important and valuable skills a player can have. We present a number of different ways to do these calculations, from a rough guesstimate system called the 2-4 Rule to the actual combination math.
The first odds calculation that must be made is to determine the total number of possible poker hands in a deck.
As we�ve shown, a poker hand consists of 5 cards drawn from a deck of 52 cards. Therefore, the number of combinations is COMBIN(52, 5) = 2,598,960.
If you use Microsoft Excel, you can duplicate these calculations using the COMBIN factor. COMBIN returns the number of 67 combinations for a given number of items. To find the COMBIN factor in Excel go to INSERT . . . FUNCTION . . . MATH & TRIG.
For each of the above ?Number of Combinations? we divide by this number to get the probability of being dealt any particular hand.
For the calculations, we will first split out the ?No Pair? hands which include Royal Straight Flushes, Straight Flushes, Flushes, Straights, and ?Nothings?. Then, we will look at all combinations that have at least 1 pair.
The cards in a hand without any pairs will have 5 different denominations selected randomly from the 13 available (2, 3, 4...Ace). Also, each of the 5 denominations will select 1 suit from the four available suits. Thus the total number of no-pair hands will equal: COMBIN(13, 5) * (COMBIN(4, 1))^5 = 1287 * 1024 = 1,317,888.
A Straight Flush is made up of 5 consecutive cards in the same suit and may have a high card of 5, 6, 7, 8, 9, 10, Jack, Queen, King, or Ace for a total of 10 different Ranks. Each of these may be in any of 4 suits. Thus there are 40 possible Straight Flushes. An Ace high Straight Flush is a Royal Flush. Since there are only 4 different suits there are only 4 possible Royal Straight Flushes. When we subtract the 4 Royal Straight Flushes from the total of 40 Straight Flushes we are left with 36 other Straight Flushes that are King high or less.
A Flush consists of any 5 of the 13 cards from a particular suit.
There are 4 possible suits. The number of possible Flushes is: COMBIN(13, 5) * 4 = 5,148. However, this includes the 40 possible 68 Straight Flushes. When we subtract these out, we are left with: 5,148 - 40 = 5,108 possible ordinary flushes.
A straight consists of 5 cards with consecutive denominations and may have a high card of 5, 6, 7, 8, 9, 10, Jack, Queen, King, or Ace for a total of 10 different Ranks. Each of these 5 cards may be in any of the 4 suits. Thus there are 10 * 4^5 = 10,240 different possible straights. However, this total includes the 40 possible Straight Flushes. Thus we subtract 40, which leaves us with 10,200 possible ordinary straights.
Finally, we come to the ?nothing? hands which are basically all the left over garbage. This is simply the total number of ?No Pair? hands minus all the good stuff. This gives us: 1,317,888 - 4 - 36 - 5,108 - 10,200 = 1,302,540 ?nothing? hands.
How about the odds of getting 1 pair or better? A hand with just 1 pair has 4 different denominations selected randomly from the 13 available denominations. 3 of these denominations will select 1 card randomly from the 4 available suits. The 4th denomination will select 2 cards from the available 4 suits. Finally, the pair can be any one of the four available denominations. Thus the calculation is: COMBIN(13, 4) * (COMBIN(4, 1))^3 * COMBIN( 4, 2) * 4 = 1,098,240 possible hands that have just one pair.
The calculation for a hand with Two Pairs is similar. We will have 3 random denominations taken from the 13 available. Two of these denominations will use 2 of the four available suits while the third denomination selects 1 of the four available suits. The Singleton card may be any one of the three denominations. Thus, the calculation becomes: COMBIN(13, 3) * (COMBIN(4, 2))^2 * COMBIN(4, 1) * 3 = 123,552 possible hands with 2 pairs.
69 Three Of A Kind is calculated in a similar manner. There will be 3 different denominations from the 13 possible denominations. One denomination will select 3 of the 4 available suits while the other two denominations select 1 card from each of the 4 possible suits.
Finally, the Three Of A Kind can be in any of the three denominations. The calculation becomes: COMBIN(13, 3) * COMBIN(4, 3) * (COMBIN(4, 1))^2 * 3 = 54,912 possible hands with 3 of a kind.
The next calculation will be for a Full House. A Full House only uses 2 of the 13 denominations. One of these will select 3 cards from the 4 available while the other selects 2 cards from the 4 available. Finally the denomination that has 3 cards can be either one of the 2 denominations that we are using. This gives us: COMBIN(13, 2) * COMBIN(4, 3) * COMBIN(4 , 2) * 2 = 3,744 possible Full Houses.
The final calculation is for 4 of a kind. Again, we will select 2 denominations from the 13 available. One of these will select 4 cards from the 4 available (Obviously the only way to do this is to take all four cards.) while the other denomination takes 1 of the available 4 cards. The denomination that has 4 of a kind can be either one of the 2 available denominations.
Thus, the calculation becomes: COMBIN(13, 2) * COMBIN( 4, 4) * COMBIN( 4, 1) * 2 = 624 different ways of being dealt 4 of a kind.
Poker Odds From The Turn Many players who really understand Hold'em odds still tend to forget that the �turn� can change their odds dramatically. It's true that for a flush draw, the card odds are 1.9 to 1 from the flop to the river. However, this is a theoretical situation where it assumes 70 there is no additional betting on the turn. Typically this is not going to be the case so you will need to recalculate your card odds and pot odds.
We will use the flush calculation example again and run through it 100 times assuming there was $20 in the pot on the flop with two $5 bets. On the turn, this leaves $30 in the pot, plus a $10 bet from your opponent to call.
Cost to Play = 100 hands * $10 to call on turn = -$1,000 Pot Value = $30 + $10 bet + $10 call Odds to Win = 4.1:1 or 19% (From the turn) Total Hands Won = 100 * Odds to Win (19%) = 19 wins Net Profit = Net Cost to Play + (Total Times Won * Pot Value) = -$1,000 + (19 * $50) = -$1,000 + $900 = -$100 Profit Now, you can see that what was a very profitable draw on the flop suddenly turned into a not so great draw on the turn. This is because by not hitting your flush by the turn, it lowered your chances of making a flush by the river. The odds thus increased to 4.1 to 1 instead of 1.9 to 1. So even though the pot odds remained the same at 4:1, because the card odds went down, this flush draw has now become unprofitable.
Realizing the dynamic changes in your odds is extremely important so that you don't go making incorrect draws based on odds from the flop. Just remember that your odds essentially double from the flop to the turn, so adjust your play accordingly.
71 Each entry in the following table is the result of 1,000,000 simulated hands of Texas Hold 'em played to the showdown and represents the percentage of pots won (including partial pots in the case of splits) by the indicated hand against the indicated number of opponents holding random hands.
The study shows a very clear correlation between your odds of success against the number of players. Notice the JJ, TT, 99 anomaly where the power of these cards increase dramatically over perceived better pocket cards - depending on how many players are left. The hands indicated in BOLD can have impressive results but require aggressive raising to force out weaker players.