Odds, probabilities and bookmaker ('s commission)

Now, from the previous article on odds & probabilities, the reciprocal mathematical relationship between odds and probabilities is established as a basis in sports betting.
As one can test (quickly and for free !) with the following simulator, this math relationship is made so that, on average over a large number of bets, we neither win nor lose: losses and gains balance out.
Mathematically speaking, this is just an illustration of the mathematical expectation being zero. Actually, the greater the number of games, the more obvious this balance is.
A large number of bets: that’s exactly what the bookmaker sees! from his point of view him, and by offering odds exactly the reciprocal of the probability of events, no gain or loss (try the simulator with 10,000 or 100,000 games).

First of all, this is disappointing. If one agree with the odds/probability of a bet, then one might as well not wager ! this will not bring anything on average in the long term. Some profits, only temporary, just for a while, a kind of illusion, and before losing again, ..., for an overall balance.

You should therefore only bet if you estimate that the implies probability given by the odds (the reciprocal of the odds therefore) underestimates the real probability (which you estimate yourself elsewhere, or with the help of tipsters). In this case, which is called a value bet, yes, on average over a certain number of games of this type, you will be a winner.
For example, if a bookmaker gives odds of 1.9 for Federer on a match, while I personally estimate that Federer has more than a half chance of winning, more precisely has for example a probability of 60% (3 chances out of 5, see also the quick conversion table), which would correspond to a odds of

c = 1/60% = 1/0.6 ≃ 1.66

then I estimate that the bookmaker underestimates the probability, and therefore overestimates the odds, and I must bet!
Actually, the situation is not only uninteresting if we consider that the odds/probability of a bet are mathematically (statistically?) correct: it is even disappointing! because the probability is in fact always over-estimated by the bookmaker who, as a salary for his work, thus takes a margin on each bet.

Bookmaker's margin and (almost) everyone becomes a loser

In reality then, secondly: it's much worse!
Indeed, to bet, we need an intermediary: the bookmaker. The latter is not a philanthropist, he will be paid for his work (being the intermediary, calculating and providing the odds, managing the bets, etc.). He gains this margin by playing on his evaluation of the probabilities and therefore the odds that he provides.

In the following, I imagine myself a bookmaker.
I am a bookmaker and I calculate the probabilities and odds of the outcomes of an event, for example

1 in 3 chance for victory, i.e. 33%, and therefore an odds of 3
2 chance out of 3 for defeat, i.e. 66%, and therefore an odds of 1.50

(see this page for explanations of the math calculations and this quick probability/odds correspondence table).
So, on average over a large number of players, I'm not going to win anything. This is when I introduce my profit margin, for example 10% margin spread over victory and defeat.

for victory, 33%−5% = 28%, et donc une cote de 1/28% = 3.57
for defeat, 66%−5% = 61%, et donc une cote de 1/61% = 1.63

and here are the new odds that I will offer to my bettors.
Why this balanced distribution of my margin, 5% / 5% ?
Depending on the odds, more bettors could be attracted by one bet than the other, and thus unbalance my (bookmaker's) winnings. It is therefore also in my interest to distribute my margin in such a way as to attract bettors towards more attractive odds! (as do my competitors for example). The distribution will therefore not necessarily be 5%/5%, but that is the work of the bookmaker, and this point interests us less at first.

Readjustment of odds at the event

Before publishing my odds to punters, I also look at my competitors, the other bookmakers. I wish more punters would come and bet with me rather than with my competitors.
I therefore readjust my odds if necessary:

in order to possibly attract more bettors to me, therefore sometimes offering, on certain events, more interesting odds than my competitors
in order not to create too great an imbalance in attractiveness between my odds: if too large a number of bettors bet on a particular event, the imbalance could be harmful to me
in order not to create a surebet between my odds and those of other bookmakers
in order to also check that I have not made a calculation error, and monitor the real current context (player who has just been injured, etc.)

Finally, I still readjust my odds gradually as punters bet, in order to keep a balance between bets and winnings and preserve my margin.

The bookmaker always wins

The bettor bets against the odds. The bookmaker bets with the odds.

The preceding elements should be quite convincing as to the expression "beat the bookmaker" or "bet like the bookmaker" or even "bet against the bookmaker".
The bookmaker always wins because he sees a large number of bettors and the gains and losses of these bettors balance out minus his margin.
To be a winner in the long term, you must not focus on a bet, but on a betting strategy such as finding value bets, using a math betting strategy or mastering martingales.