In most places related to sports and betting you will see the odds of a team winning rather than the probability.

The reason you will see odds instead of probability, is because the odds tells you how much you win if you placed a bet on the given odds.

E.g. if you bet $100 @ 2.00 in odds, you would get $200 in return if you won.

One could have used probability instead to describe how likely it was that you would win the bet.

But, then it would be less clear on how much you’d win if you bet $100. The relationship between odds and probability is that odds = the inverse of the probability (odds = 1 / probability).

If one is betting on the true margin of a bet with 2 outcomes, e.g. a coin flip, you would expect to break even in the long run, no matter which side of the bet you chose. Let’s use an example to show that this is true.

We will use the example with Andrew and Bobby. Andrew (A) wants to bet with Bobby (B) on whether he can hit the crossbar or not. Historically, he hits the crossbar 4 out of 5 times and misses 1 out of 5 times. This would give the corresponding odds of 1/(4/5) =

1.25 for a hitand 1/(1/5) =5 for a miss. Andrew wants to bet $100 on himself hitting the bar, so he then asks Bobby what odds he can give him. Bobby knows the math so he says he can give him odds of 1.25.

Andrew has now bet $100 on the odds of 1.25, giving him a potential return of $125, or a profit of $25.

In a sample size of only 5 kicks to hit the crossbar, this would generally not happen because the variance (or standard deviation) is too high.

But, when you do this over a large sample size, say he is kicking the ball 100,000 times, the variance becomes smaller and the probability that is reflected in the odds correlates much better with the probability for the real events.

If this is confusing you should read more about the law of large numbers.

**It is important to notice that they would break even only for events where the odds and probability relates to each other e.g. a coin flip.**

Bobby now gets a downgrade in IQ, so he does not understand the math.

Andrew then asks again, what odds Bobby can provide if he wants to bet $100. Bobby says he would give him **1.5 for hitting the crossbar**, while **giving himself the odds of 3**.:

But, if the odds were to relate to the probability, we would give the odds of 1.25 and 5, as in the last example. Let’s see what happens with 1.5 and 3 in odds:

Andrew still bets $100, with a potential return of $150, or a profit of $50.

Bobby is betting $50, with a potential return of $150, or a profit of $100.

As we can see, A wins, while B loses in the long term. This is because the odds of him hitting the bar is overvalued (or the underlying probability is undervalued) while the odds of him missing is undervalued (or the underlying probability is overvalued).

When the probability is undervalued it means that the event happen **more often than the probability suggests in the odds**.

But then you might want to ask: But no sports match is the same, so how is this applicable to the real world?

Well, over a vast amount of games there is large evidence for the probability to be reflected in the odds. If you want to read more about it, check this article out.

Odds and probability are **in theory** really just two sides of the same coin, but **in practice** the case is different.

This is because (soft) bookmakers usually offer unfair odds, which do not reflect the underlying probability.

This is the way the bookmakers make money, so it is important to understand that **the odds at the bookmakers almost never reflect the underlying probability.**

We have written articles explaining them:

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