In probability there are two common and different ways to look at the possible outcome of an event. One is to express the mathematical odds, such as 4:1, and the other is to express the chance, such as 20%. It is important that you can easily switch between these two forms as both are used, and sometimes one is easier to calculate or easier to use in a specific situation.

Odds

Odds are simply the measure of all the undesired outcomes to the desired outcome. It is a direct relation between those two sets of outcomes.

If you wish to know your odds of selecting a heart from a deck of cards you simply need to count all the outcomes. There are 52 cards, 13 of which are spades, thus 39 of which are not spades. Your odds to select a spade are 13:39, which can be reduced to 1:3.

It is important to note that this 1:3 are the odds that you do you draw. This may seem confusing since you are used to the 3:1 notation. 3:1 is actually your odds that you don't draw a spade. The exact form in which the odds are expressed depends on what question you are asking.

Payout Odds

This however doesn't make any sense if you are talking about something like pot odds. When somebody says you have 4:1 pot odds what they actually mean is that for every $1 you put in, you'll get $4 out if you win.

You can see that this has no relation to the actual chance of you winning the hand. It is however a very useful form when comparing to other true odds, and it can be converted just like any odds.

Note that most formulas don't deal with percentages directly and express them instead as a decimal value between 0 and 1. This is a straight forward division of multiplication by 100. For example, 50% = 0.5 and 7% = 0.07.

Percentage

The common display of a percentage number is the probability that a desired event will occur. It is a single value expressed in relation to all outcomes.

Using the same example, what is your chance, or the probability that you will select a spade from a deck of cards. In this you have a 25% of drawing a spade.

Converting

The formula relating odds and probability is o = p / (1-p) where o=Odds and p=Probability. The division is not performed and instead replaced with a : to express the odds, such as 1:3. Also note that 1 refers to the total outcome space.

In our first example, to pick a spade from a deck, we thus have o = 13 / (52 - 13). Notice that we have replaced the 1 with 52, since that is the total number of outcomes. So we end up with o = 13 / 39 = 1 / 3 = 1:3.

Inverting the formula we see then that our probability is p = o / (1 + o). In this case we treat our odds as a normal number (replace the : again with a division). The probability to select a spade is then p = (1/3) / (1 + 1/3) which is p = 0.25 = 25%.

That may seem hard to do in your head, but it can be simplified. If you odds are expressed as x:y then you probability is simply x / (x+y), simply divide your good outcomes by the sum of the total outcomes. The sample example again is thus simply p = 1 / (1+3) = 25%.

Terminology Warning

The example used above illustrates a possible cause for confusion. While odds and probability are in the end just numbers, the form in which they are expressed is important, and you can't afford to get it wrong. This is of course complicated by using words to express what you mean.

For example, your odds to select a spade are 1:3. The probability that you will select a spade is 25%. The probability can also be stated as a 1 in 4 chance. Additionally your probability can be expressed as the fraction 1/4. Not replacing the / symbol your odds can also be expressed as 1/3. And strictly speaking you odds could also be reduced to a single number 0.33.

Therefore the convention in poker, or gambling, is to express probability as a % value and odds as two colon separated numbers. As you can see however it is vital you understand the difference.