Odds and Probability

Recently, I read a prepublication version of a hold'em book written by Bill Barnes. Bill is rather fussy about proper usage of the terms ``odds'' and ``probability''. Several writers who have made errors when writing about odds have been on the receiving end of friendly critiques from Bill.

Frequently, I read statements on the Internet in which the message writer uses the term ``odds'' in a way which would make Bill wince. Let's talk about the proper use of the term ``odds'', and the context in which it makes sense to use it.

Typically, we are interested in some outcome A. For example, if we are holding two hearts in the hole while playing hold'em and there are two hearts on board after the turn card, we are interested in the outcome that the river card is another heart.

The outcome A is made up of equally likely individual events (this need not always be the case, but for our purposes it is sufficient to consider equally likely events). In the previous example, there are 46 unseen cards and it's equally likely that any one of them may be the river card. So, we think of the occurrence of any single card as an event. Since there are nine unseen hearts, A consists of nine events.

Now let B denote the outcome that A does not occur, so the number of events in A plus the number of events in B, denoted |A| and |B|, respectively, add up to the total number of events.

In the example of the flush draw, |A| = 9, |B| = 37, and |A| + |B| = 46.

Note that there are three numbers floating around: the number of ways an outcome may occur, the number of ways the outcome does not occur, and the total number of events. The terms ``odds'' and ``probability'' are defined using ratios involving the three numbers.

We define the probability of A occurring to be |A|/(|A|+|B|). We define the odds against A occurring to be the ratio |B|:|A|, and usually write it as |B|-to-|A|. We define the odds for A occurring to be the ratio |A|:|B|, and usually write it as |A|-to-|B|.

Applying the preceding definitions to the flush draw example, we obtain 9/46 as the probability of the player making a heart flush, and 37-to-9 as the odds against the player making a heart flush.

There are times when it is appropriate to use ``probability'', and times when it's appropriate to use ``odds''. Let's look at a few examples.

Baseball batting averages are given as probabilities. A batting average is obtained by dividing the number of hits by the total number of official at-bats. Suppose Ichiro's batting average is .333 (at the time of this writing, it is much higher). This means he is getting a hit once in every three times at bat and that's the way we always think about it. If you were to tell a baseball fan the odds against Ichiro getting a hit are about 2-to-1, you would be looked at in a quizzical manner. It simply does not convey the same information as saying his chances of getting a hit are about one in three.

You might argue we are so accustomed to thinking of batting averages in the way we do that using odds fails for that reason. I would disagree and say it is more natural to use probability for batting averages. The reason for this is that the two numbers we have staring us in the face are total at-bats and number of hits. We have to do some work to obtain the number of times the player didn't get a hit, and the latter number is what we need to express odds.

The preceding example demonstrates a situation for which probability is the natural concept to use. Let's now examine a situation where odds is the natural concept to use.

If there is $50 in the pot and you are deciding whether or not to call a $10 bet, the numbers you have in hand are 10 and 50. Suppose you are certain to win the pot if an outcome A occurs, but certain to lose the pot if A does not occur. If the probability of A occurring is 1/6, then you win $50 one out of six times and lose $10 five out of six times. This means you are breaking even in the long run by making the call.

Thus, a probability of 1/6 is the break-even point for making a $10 bet with $50 in the pot. The relationship between 1/6, 10 and 50 is not particularly nice expressd this way. However, note that the odds against A occurring are 5-to-1 when the probability of A occurring is 1/6. Now the ratio 5-to-1 is precisely the same as the ratio 50-to-10. That is very striking and you see the beautiful fact that the break-even point for a bet is exactly when the ratio of the pot size to your bet is equal to the odds against whatever outcome you need to win.

Real games are not always as clean-cut as the preceding example. The point, though, of the example is to illustrate that you should be thinking about odds rather than probability when contemplating whether or not to bet. The ratio of the pot size to your required bet is an important enough concept that it has a name: ``pot odds''.

Many poker players intuitively make proper moves even though they have not thought one bit about the underlying odds. When you hear a player say, ``There's too much money in the pot. I have to call,'' there is a good chance he is correct. He probably has pot odds working in his favor.

Let me finish with an example which, though not actual, illustrates the sloppy usage of the term ``odds'' I am seeing all too frequently on the Internet: ``What are the odds of being dealt a full house in draw poker?''

Where is the sloppiness in the question?

When using the term ``odds'', you should say either ``odds for'' or ``odds against''. The fact the question poser wrote ``odds of'' indicates he probably was thinking of ``chances of'' or ``probability of''. If that is the case, then we have to wonder whether or not he actually understands odds.

Furthermore, there are contexts in which it would be more useful to know the probability of a full house occurring rather than the odds against a full house occurring. When the question is sloppily posed, we are not certain the form the answer should take.

English is a dynamic spoken language. Colloquial usage eventually prevails over codification. Nevertheless, I am trying to make a small contribution in the fight against people using ``odds'' to mean ``chances'' or ``probability'' in colloquial English.

The odds against winning this fight are daunting.