Odds and probabilities

Maths underlying odds and gambling are the basis to understand as they can help determine whether a wager is worth pursuing
It is actually highly important to understand the link between the odds of a particular sport event, which are link to the total payout for an exact pronostic, and the probability of this success occurring, that is to say the mathematical relationship between what I can potentially earn and the chances I have of earning it.
Mathematical expectation is the fundamental calculation tool for all random games, therefore for all bettors, traders, but also on the other side for the bookmaker who computes and offers the odds of the event.

Basic Heads or Tails – Balanced Bet

Let's take the basic, but fundamental, example of a well-balanced coin. I bet say 1 euro. If the result is Tail, I earn 2 euros otherwise I lose my wager and we leave with 0 euros.

	Tails	Heads
Payout (euros)	2	0
Probability	50%	50%

and expectation is then the mathematical computation of the probability-weighted average:

E = 2×50% + 0×50% = 1

This game is balanced: on average, over a large number of such games, we neither earn nor lose anything.

Odds are the number by which the stake invested is multiplied when the wagerer wins his bet. Here we have odds of 2: I bet 1 euro and earn 1×2 for an exact pronostic.

Heads or tails loaded - Unbalanced bet

We slightly modify the previous game. Let's say that for now the coin comes up Tails in 40% of cases. If we keep the same rule as before: stake are doubled when I win whereas I still simply lose my wage when I miss my pronostic, so that we have

	Tails	Heads
Payout (euros)	2	0
Probability	40%	60%

and the mathematical expectation is now

E = 2×40% + 0×60% = 0,8

which is now lower than the stake: on average, this game is unfavorable to the player who will lose some 20 cents per game (on average over a large number of such games). The odds of 2 for victory (Heads) are not enough to balance for its lower probability.

What odds are to be applied in order to rebalance this game? Denoting c such that

	Tails	Heads
Payout (euros)	c	0
Probability	40%	60%

We simply calculate the expectation: E = c×40% + 0×60% = c×40%, which shows that we must have, to balance this game (therefore hope to balance the stake of 1 euro):

E = c×40% = 1 ⇔ c = 1/40% = 2,5

You must therefore have odds of 2.5 for this game to be balanced, according to these probabilities of winning and losing.

Here we have the very definition of odds and which must absolutely be kept in mind: odds are calculated so that the game is mathematically, in probability, balanced, with this correctly calculated odds, on average there is neither profit nor loss ! (remains to think about the bookmaker's margin, which will reduce this balance towards the general loss … see after )

General case: odds / probability / payout relationship

The general game looks like this. I bet m with odds c: in the event of victory I earn m×c, and in the other case I lose my wager.
Probability of winning is p, therefor 1−p for defeat (0,4 et 1−04 = 0,6 dans l'exemple précédent).

	Tails	Heads
Payout	m×c	0
Probabilité	p	1−p

The expectation is therefore E = m×c×p + 0 and for this game to be balanced, this expectation must therefore be equal to the stake, that is E = m×c×p = m and which gives us the relation fundamentals of betting

c = 1/p

or,

The odds are the reciprocal of the probability of winning

This relationship is very commonly cited. Here is its explanation and above all its consequence for the player and wagerer: if the odds are correctly calculated by the bookmaker, it does not allow (on average) to obtain any profit.
Simulator and calculator below illustrates this fact. This fact is illustrated by the simulator below.

Gain Simulator / calculator

With this simulator / calculator, either the odds or the probability can be filled, and the other is computed automatically.
As many games as desired can be simulated. Here, stakes are 1 euro per game, and so the odds give directly the potential payout.
For a small number of games, one can hope to win (or lose!). On the other hand, the more the number of games increases, the more the gain clearly stabilizes around the expectation (try with 1000 or 10000 games for example). Imagine for a moment if the expectation was clearly, from the start, smaller than the total stake! And that's exactly what actually happens... with the bookmaker's commission …

Odds/probability relationships to know

To make an informed decision, to estimate your chances, to flush out a possible value bet or sure bet, … one must know how to quickly convert, without a calculator or other tool, the probability into odds and chances of winning.
The following table gives this mathematical correspondence, or translation:

Chances	Probabilities	Odds
1 in 5 chance	20%	5.00
1 in 4 chance	25%	4.00
1 in 3 chance	33%	3.00
1 in 2 chance	50%	2.00
2 chance out of 3	66%	1.50
3 chance out of 4	75%	1.33
4 chance out of 5	80%	1.25