Is there any advantage in using odds in probability ad statistics?

Question

The question title says it all. I have always wondered why in some areas, mostly gambling, people use odds instead of probability. But I have never seen a probability/statistics book (at least one directed to mathematicians) that used odds.

Is there any advantage in using odds or is it just a historical habit?

The only thing I could come up with is that it may be simpler to understand odds than probabilities. For example, the odds against obtaining 6 when rolling a fair dice is 5:1, so we have 5 times more chances of losing than winning.

But the probability of winning is $\frac{1}{6}$, so we would expect to win once out of $6$ attempts. Not really harder to understand intuitively.

Also, odds don't seem that easy to handle when a random experience has more than 2 outcomes.

Answer 1

• If you give me the probability $p$ of an event, I can give you the odds against it by computing $t = \frac{1-p}{p}$.

• If you give me the odds against an event as real-number ratio $t$, I can give you the odds by computing $p = \frac{1}{t+1}$.

Hence, the two are equivalent. Any advantage would be one of convenience or tradition, although several such instances exist. For instance, the statement that probabilities are additive for disjoint events is certainly much more conveniently expressed about probabilities than for odds; similarly, as K B Dave alluded to, logistic regression is a good example of an application that natively uses odds. But at the end of the day, you can always pick your favorite and convert to it.

I think the reason odds are prevalent in gambling is some combination of history/tradition and (perceived) ease of interpretation.

Answer 2

Bayes' Theorem can be expressed succinctly in terms of odds. This is called Bayes' Rule: $$O(A_1 : A_2 \;| B) = O(A_1 : A_2) \cdot \frac{P(B|A_1)}{P(B|A_2)}$$ which can be stated

posterior odds = prior odds x likelihood ratio

The factor $$ \frac{P(B|A_1)}{P(B|A_2)}$$ is called the likelihood ratio or the Bayes factor.

Answer 3

The obvious reason for their prevalence in gambling is that odds arise naturally as ratios of stakes. If I give you odds of $5:1$ against rolling a six, I simply put in $\$5$ for every $\$1$ you do, and then the winner takes the pot. And to look at the same bet from the other person's point of view, you just flip the odds.

They're not so convenient mathematically, so mathematicians use probabilities. But gambling has a much longer history than (and inspired) the mathematical study of probability.