Odds and Probabilities
Written by John Porter
Thursday, January 19, 2006
Odds and probabilities are important to the game of poker, they require a little bit of math, but they are really not that difficult once you practice with them. The key is to spend some time working with them so you get comfortable in using them, and understand what is going on.
Why are Odds and Probabilities Important
Why must we understand odds and probabilities? As I explained in another article, the skill part of poker, to a large extent, consists of making good bets and avoiding bad bets. The player who makes the best decisions will win in the long term. Make the wrong decisions, and you will lose over the long term.
In a coin flipping example, there is an equal chance of getting a head or a tail. If you were to put up a dollar for every head that turned up, and your friend put up a dollar for every tail that turned up, you would have an even bet. Since the chances of a head or tail on any coin flip are equal, your odds are also 1 to 1. One time you will win, and one time you will lose. In this example, both you and your friend expect to win the same amount over time, $0. This is neither a good bet nor a bad bet, but neutral. You shouldn't be interested in taking the bet however because the best you can do over time is break even.
On the other hand, if you put up $1 for every head, and your friend put up $2 for every tail, you would have an edge. The chances of flipping heads and tails will still be even, but you get paid more when you win, so over time you will make a profit. This is a bet you should take since you expect to make a profit over time. If you had to put up $2, and your friend put up only $1, you would expect to lose money. This would be a bad bet for you.
In a coin flipping example, the choices are easy to understand, since there are only 2 possibilities. This is not always the case. In poker the choices are much more complicated, which is why it is important to understand odds and probabilities, in order to make good decisions.
Probability
Probability is the likelihood that something will happen. For instance, when you hear the weather report in the morning, and the weatherperson tells you that there is a 20% chance of rain they are saying that the probability of rain is 20%.
Some important concepts to understand here are that if there is a 20% probability that it will rain, there is an 80% probability that it will not rain. Probabilities can not add up to more than 100%, and the sum of all of the various possibilities must add up to 100%.
In simple cases like a coin flip, or the chance of rain, where there are only 2 possibilities, the 2 probabilities will add to 100%. In some situations however there will be more than 2 possibilities. If we only calculate some of the probabilities, they will not add up to 100%, because we did not consider all of the possibilities, but those possibilities still exist, and must add up to 100% in total.
Another way to write the same information is to say that there is a .2 probability of rain, and that there is therefore a .8 probability that it will not rain. The total probability can not be more than 1, and once again all of the possibilities must add up to 1.
Odds
Odds are a different way of expressing the same information, but in a way that is often more applicable to poker and other gambling games.
While probability is expressed as a decimal number, or a percentage, odds are expressed as 2 numbers separated by a colon such as 5:1. By convention this notation indicates that the odds are 5 to 1 against the event occurring.
There are different ways of saying the same thing, and of explaining what the numbers mean. In the example, let's assume that the event we are interested in is getting 1 particular card that we need in order to make our hand. The notation tells us that 5 times we will fail to get the card we need, and 1 time, we will get the card we need. Using that same example, we will get the card we need 1 time in 6 attempts, or 1/6.
Working with Odds and Probability
Note that although probability is normally stated as a percentage, or a decimal number, percentages and decimal numbers are simply fractions expressed, or written, in a different way. For instance, 1/6 is the probability of getting the card we need. If you divide the 1 on top, by the 6 on the bottom, you get .167, or 16.7%. All 3 of these numbers mean exactly the same thing, there is a probability of 1/6, or .167, or 16.7% of getting the card we need.
Putting it all together, 5:1 means losing 5 times for every 1 win, winning 1 time out of 6 attempts, the probability of getting the 1 card is 1/6, .167 or 16.7%. The probability of not getting the card you want is 1  .167, or .833, or 83.3%. Once you know the probability of getting the card, and the probability of not getting the card, you can put that information into the form of odds. In our example that becomes 83.3:16.7 against getting your card.
You normally reduce odds to the form X:1 to make comparisons easier. To do that, you simply divide both numbers by the number on the right. i.e. in the example 83.3:16.7 you divide 16.7 by 16.7 to get 1, and then divide 83.3 by 16.7 to get 5, giving you 5:1, which is exactly where we started.
Of course if you do the math you will see that I rounded the number off in all cases since numbers like .16666666666 are difficult to work with, and for our purposes, .167, .833 and 5 are plenty accurate enough.
Going back to the weather example from the beginning, there is a 20% chance of rain, which means that there is an 80% chance that it will not rain. Putting these numbers in the form of odds, it is 80:20 against it raining. Simplifying, divide both sides by 20 and you get 4:1 against it raining. You can put this back into the form of a probability by adding the 2 numbers together and then putting the right number on top, i.e. 4 plus 1 is 5, put the 1 from the right side on top of that and you get 1/5. There is one chance in 5 that it will rain. To express the fraction as a decimal number, divide the number on top by the number on the bottom, i.e. 1 divided by 5 and you get .2. To express that as a percentage, multiply by 100 and you get 20% chance of rain. Right back with the number we started with, because they are all ways of saying the same thing.
Why use Both
Stating the situation in the form of odds, as in 5:1 gives us a clearer picture of where we stand than saying we have a 16.7% chance of getting the card. As well, it gives a more complete picture since, for the probability we want to know both that there is a 16.7% chance of getting the card and an 83.3% chance of not getting the card.
Odds can not be used in all situations however. For instance, on the first card you are dealt, the odds of getting an Ace are 12:1, the odds of getting an Ace on the second card, given that you got an Ace on the first, are 16:1. If you want to know the odds of getting a pair of Aces however, you can not calculate them directly from the odds, you must use probabilities.
Using probabilities to do this, there are 4 Aces out of 52 cards, so the probability of getting an Ace on the first card is 4/52 or 1/13. The chances of getting an Ace on the second card are 3 Aces, since we already have 1, in 51 remaining cards, which is 3/51 or 1/17. You can then multiply the 2 probabilities to get the answer.
You can do this in 1 of 2 ways. You can multiply the fractions 4/52 * 3/51, or 1/13 * 1/17, to get 12/2652 or 1/221 and then convert to odds. i.e 265212:12, is 2640:12 is 220:1, or 2211:1 is 220:1.
You can also convert each of the fractions to decimals, 4/52 ~ .077, and 3/51 ~ .059, then multiply .077 * .059 ~ .0045, convert to a percentage by multiplying this number by 100 and there is a .4525% of getting a pair of Aces as your first 2 cards. Since there is a .4525% chance of getting a pair of Aces, there is a 100  .4525 = 99.5475% chance of not getting a pair of Aces. The odds against getting a pair of Aces on the first 2 cards are 99.5475:.4525, simplifying, we divide both sides by .4525 and we end up with 220:1, the same answer.
Note that when doing a series of operations such as above, you can't round the numbers off until you complete all of the calculations or it will significantly affect your results. I used numbers such as .077 above instead of typing out the entire long decimal number, but I used the actual numbers in the calculations.
Of course, trying to do this math at the table would not be practical, so for many common situations we memorize the odds, or probabilities. For instance, the odds of someone having any 1 specific pair as their first 2 cards are 220:1. i.e. it is 220:1 that they will have KK, 220:1 that they will have QQ etc. In order to make memorizing easier, I will provide tables of many common odds and probabilities in later articles.
As we will see in the next Odds related article, there are a couple of more good reasons to use odds instead of probability. One is that odds are much easier to calculate while sitting at the table. The other is that the odds can be used directly in deciding if we have a good bet or a bad bet.
Contact the author at http://A1PokerInfo.com
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