Odds

Calculation Odds
Definitions:
A possibility is an event or outcome that can occur or happen in a particular circumstance.
A probability is the likelihood that a possibility can occur. A probability is often expresses as a percentage , a ratio (say favourable outcomes to unfavourable outcomes i.e. odds) or the number of favourable outcomes in a set of outcomes.
If an event or outcome has say a 10% probability of occurring, the odds are 1:9 (one to nine) it will occur or stated differently, it will only occur 1 time in 10 occurrences. Put differently, the event while possible has a low probability or chance of occurring, or the outcome has a small chance of happening or a good chance of it not happening.
A 100% probability that an outcome can occur, is a certainty it will occur. In poker, the certainty of having no outs (the number of cards left in the deck that will improve a hand) is called drawing dead. Having a winning combination of cards that has a 100% probability of winning a hand is called the nuts.
Odds of a hand are the probability of making a hand vs. the probability of not making a hand.
Pot odds are the amount in the pot compared to the cost of making a bet. If the minimum bet required to continue playing is $1 when there are $10 in the pot, the pot odds are 10:1 - in other words, if the players wins the pot, the player will make a ten-fold return on the bet or the players wins ten times the bet. However, even though the odds look attractive  if a hand is drawing dead, there will be no return on the bet if the hand proceeds to a showdown (unless the player with the nuts mucks her or his hand because of, say, a bluff). Assuming, a hand will proceed to a showdown, pot odds also need to consider the odds of making a hand.

How to Convert a Ratio to a Percentage
Example: Covert 4:1 to %
Add two sides of a ratio: 4 + 1 = 5
In the ratio, 4 = 4 ÷ 5 x 100 = 80% (say, 80% chance of an outcome)
In the ratio, 1 = 1 ÷ 5 x 100 = 20% (say, 20% chance of an outcome)
Therefore if the odds of winning to losing are 4:1 = the chance of winning is 80% while the chance of losing is 20%.

Calculating Odds of Making a Hand
Calculating odds is calculating the probability of making a desired hand or a range of possible winning hands. Often, calculating odds is not an exact science and is riddled with assumptions. The quick calculations this writer uses is at best a rough personal guide and is different from those suggested on other sites and which this writer finds too complicated. Hands with high odds can lose and hands with low odds can win, which is why playing poker is called gambling. Some of the steps this writer uses in calculating odds are:

1. Count Outs
There are several ways of calculating the odds of dealt a desired card (an out) of which two are as follows:

• Compare the number of outs to the number of cards remaining in the pack (the examples below). Comparisons are expressed as a ration.
• Compare the number of outs to the number of unknown cards (in the dealer's hand, in the opponents' hands and any cards burnt by the dealer).

In our method, the number of cards (called outs) needed to make the hand are counted. Call this number x. For example: to make three-of-a kind (a set) with a pair in hand, the total number of cards of that rank in a pack = 4. The known cards already dealt to make the existing pair = 2. Therefore, there are 4-2=2 cards remaining that can make a set. If none of those cards have been dealt to another player, there are 2 outs to make a set. Here, x=outs=2. If another player has been dealt one or both of the required cards, the number of outs decreases to 1 and 0 respectively.

Because a player must make assumptions that can be false, realistically, a player needs to consider all options which will give a range of possibilities that can help a player decide whether or not to gamble.

2. Count Cards Left in the Deck
There are 52 cards in a full deck of cards. The number of cards left in the deck = 52 - number of cards dealt. Call this number y.

Continuing our previous example: in a heads-up game with two players, the number of cards left in the deck are:
1. Pre-flop = 2 players + 2 cards cards dealt to each player = 4 cards. The number of cards left in the deck = 52 - 4 = 48;
2. After the flop of 3 cards = 48 - 3 = 45;
3. After the turn (1 card) = 45 - 1 = 44;
4. After the river (1 card) = 44 - 1 = 43.

The calculation is complicated if a dealer burns (discards) cards from the top of the deck before placing the flop, turn or river cards on the table. One quick-n-dirty option in the calculation above is to the reduce the number of cards remaining in the deck by the number of cards burnt (discarded).

3. Calculate Odds
The odds or probability (as a percentage) of being dealt the required card = outs divided by remaining deck cards multiplied by 100 or x ÷ y.

In our example continued from above, with 2 outs, the odds of being dealt a card to make a set of three cards with an existing pair are:
1. After the turn/on the river 2 outs ÷ 44 cards remaining in the deck x 100 = 0.045 x 100 = 4.5%. Conversely, the odds of not being dealt a card on the river that can make a set is 100 - 4.5 = 94.5%.
2. After the flop/on the turn 2 ÷ 45 x 100 = 4.4% on the turn + 4.5% (from #1 above) = 9% approximately on the turn and river. [pokerroom.com states that "the chance of making trips with a flopped pair, either on the turn or the river, is 8.4%." Wikipedia also puts it at 8.4% (see below)]
3. Before the flop (this is an imprecise quick-n-dirty calculation) 2 ÷ 48 x 3 (cards dealt on the flop) x 100 = 12.5% on the flop and + 9% (from #2 above) = 21.5% from the flop to the river.

A 12.5% probability or chance of being dealt a set on the flop assuming all remaining cards are in the deck are fairly low odds but one which some players take because of favourable (in their estimation) pot odds.

Some calculations add back the opponent's cards dealt, ignore the burnt cards. In doing so, they replace 'cards left in the deck' with cards not dealt to an individual player or 'unseen cards'. Using this method:
#1 (odds on the river) above changes to 4.35% (22:1)
#2 (odds on the turn) above changes to 4.26% (22.5:1) with the cumulative % being 8.42%  (10.9:1) (taken from Wikipedia).

Other calculations subtract the number of outs from the 'unseen cards' which gives a ratio of the chance of getting the out vs. the chance of getting one of the other cards. The method used depends on what the player is trying to measure.

Comparing Pot Odds & Hand Odds
In the first example above, the minimum bet required to continue playing is $1 when there are $10 in the pot. In other words, there is an opportunity to win $10 by making a $1 bet (a 1000% return if successful) giving 'pot odds' - the odds offered by the pot - of 10:1.

If the odds on the river from 3.1 above are 4.5%, the odds expressed as a ratio are 1 to [(100 - 4.5%) ÷ 4.5] =  1 to [96.5 ÷ 4.5] = 1 to 21.4. The odds of making the hand are 1:21. In other words, there is 1 chance in 22 of making the hand (4.5% ) or 21 chances in 22 of not making the hand. [Some people express this ratio in the opposite way i.e. 21:1.]

Therefore, there is only a 4.5% chance or 1 in 22 chance of making a 1000% return on the bet. There are different ways to go about this calculation. In calculating pot odds, some people add the amount a person expects to bet to the present pot total. Others add what the opponent is expected to bet giving what are called implied odds.

In the example above, if the hand odds were, say, 20% instead of 4.5% i.e. 4:1, then theoretically, if this hand were to be played 5 times, one could expect to win the $10 pot once while at the same time making four, $1 bets - $4 in total - a 250% return on investment and an attractive proposition in a game of chance. The rule of thumb is that if the pot odds are greater than card odds, the prospect of winning a bet are in positive territory. If the pot odds are less than card odds, the prospect of winning a bet are in negative territory.

The success of this rule of thumb relies on repeated and frequent plays of this type rather than a single play winning the pot. Over time, the probability statistics indicate that the amount won from the pot should exceed the amount spent as bets - making the game profitable in the long run.

The Rule of 2
Given that calculating odds is imprecise anyway, some people use the quick rule of 2. Since there are 52 cards in a deck, each card constitutes about 2% of the deck's cards. If the number of cards left in the deck reduces to around 35 to 30, then each card constitutes about 3% of the remaining deck.

Using the rule of 2, if the number of number of outs is 2, there is a 2 x 2 = 4% chance of getting the card with each card dealt on the flop, turn or river. Since 3 cards are dealt on the flop, there is very roughly a 12% chance of seeing the card on the flop. These numbers are close to what we calculated above.

The Rule of 4
After the flop, the rule of 4 can be used to calculate the chance of seeing a card on the turn or river. Each out has a 4% chance of being seen on the turn or river. If it is not dealt on the turn, then the chance drops down to 2% for the river.