An omniscient bookmaker who gets all probabilities spot on cannot be beaten (in the long run). For this, they need to know the probabilities. In order to do so they have to set the odds accordingly. What we have seen above is that bookmakers make a profit by controlling the payout. However, the discussion generalizes to other sports too. In the remainder of this blog post we will focus on the specific game of darts, where games are head-to-head and results depend largely on the players’ skills. Note that these odds also corresponds to equal winning probabilites for each player, namely P₁ = O₂/(O₁ + O₂)=0.5 and P₂ = O₁/(O₁ + O₂)=0.5. So we see the odds the bookmaker set are not fair, fair odds would have been O₁ = O₂ = 2.0. the ‘payout’ the bookmaker sets for this game is 95%, meaning that the bookmaker will expect to make a profit of 5% over all bets, assuming they assessed the win probabilities correctly. So in the long run, each dollar spent results in 95 cents return, and you will make a loss! I.e. But what is the expected value of your return, X? Well if you bet $1 on a win for player 1, your expected return for this game is (remember that the win probability for each player is 50%): Meaning that for each $1 bet you get back $1.90 if you win. For this particular game O₁ = O₂ = 1.90 would be reasonable odds. The bookmaker can set the odds, which we will define as O₁ and O₂ for player 1 and 2, respectively. Lets take two darts players who are equally skilled and thus objectively would both have a 50% chance of winning a head-to-head game. The way they do this is by controlling what is called the payout. Both the bettor and the bookmaker can be equally skilled in predicting the outcome of a match, however the bookmaker sets the rules for the bet and thereby guarantee themselves a profit in the long run. Therefore, if you visit the casino, most likely you will make a net loss in the long run. In other words, assuming the betted amount is 1, the house has an expected return higher than 1, in contrast to the gambler who has an expected return smaller than 1. the casino or the bookmaker) has a statistical advantage. It reflects the fact that in most games of chance the house (e.g. “The dealer always wins” is a typical saying in gambling.
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