I have been thinking recently about this topic, and with my wife we started an experiment that turned out to be interesting. We managed to do, but only for a short term.

The historical average return from the S&P500 is around 8%. Official data website

In order to achieve a return like this, you only have to bet one, with your yearly investment pool on one 1.08 odds bet. If you win, you already have the average return. Anything above is a plus. But does it really worth it, how the odds are calculated and how the just one more paradox comes in play here? Let’s find out.

How are the odds calculated?

The first thing to understand is that there are three distinct types of odds: fractional, decimal, and moneyline (or American). We will focus on decimal odds in our example, but you can find a conversion table here [link to conversion table].

Using an example of decimal odds, if a team has a 2.50 odds to win its next game, the implied probability is

(1/2.50) * 100 = 40%.

The lower an odds is, the higher chance you might have to win.

Remember, as bets are coming in, the odds are changing. Thus, odds on display are never reflect the true probability or chance of an event occurring. On top of this, every bookmaker adds its own safety margin to the game.

Let’s return to the above-mentioned example. Let’s say, the other team’s odds is 1.60. Doing the same calculations as above, the implied probability is

(1/1.60) * 100 = 62.5%.

If you add up the implied probabilities, you will see that it is 102.5%. But an even happening or not happening should be 100% always, shouldn’t it? Not really. The bookmakers have to win despite the outcome. If you’d bet on both outcomes, you’d lose about 2.5% of your original stake.

The just one more paradox

The idea here is the following: when each consequent bet promises a good outcome, but the overall result will rarely be in your favor.

Let’s take the following example:

For a given event, if you win with your bet, you win 80% (1.8 odds) if you lose, you lose 50% (0.5 odds, even though it is not possible with bookmakers). For one throw 80% gain and a 50% loss means that for one throw half the time we end with $1.8 and half the time with $0.5, this means on average we end with (1.8+0.5)/2= $1.15. This is 15% more than we started with.

And even though the average result is more than the starting amount you had, the mean tends to go downward.

Example

 f: fraction of bet
 b: odds (* amount you will win)
 a: loss (100% on simple sports betting)
 p: probabilty of win
 q: probabilty of lose
 r: result

So, let’s say we start with a 100$. We put our whole pool to one 1.10 odds. The probability of a 1.10 odds to favor you is

p =(1/1.10) * 100 = 90.90%

q = 1-p = 9.1%

If we would do multiple bets with this strategy, the following equation can be written:

…to be continued