Understanding How Bookies Think About Risk

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Suppose you are given a choice between the following two opportunities. What should you do?

1. Guaranteed $500 in cash
2. A box contains five green balls and five red balls. A ball is randomly selected from the box. If the ball is green, you win $4,000. If it is red, you lose $2,500.

If you are risk averse, you would probably take the $500. After all, it's a sure thing. On the other hand, if you like to gamble, you might go for opportunity 2: Even though you could lose $2,500, your average winnings equal $4,000x.5 – $2,500x.5 = $750. This is more than the guaranteed $500, but you are taking a risk.

Let's change the game a bit. Suppose that you are now offered the same deal, except this time you don't know how many balls of each color are in the box. What should you do?

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Unless you're a compulsive gambler, you should probably either take the $500 or try to learn something about the distribution of balls in the box before making your decision.

Not knowing the distribution of balls in the box is like sports betting. For example, betting on a baseball game is different from betting on a coin toss because there is no way to precisely determine the probability that a team will win. In this situation, you must estimate the probability that a team will win (distribution of balls in the box) and compare this with the betting odds to see if a good wagering opportunity presents itself.

The Bookmaker's Perspective

Now that you've got the player's perspective, let's consider this from another angle. The bookmaker is the other side of this bet and they don't have to worry about which team wins if he can properly balance the betting action. In other words, the bookmaker has a shot at a sure thing, like the guaranteed cash in opportunity 1. Or does he?

Consider baseball money lines. Suppose the bookmaker uses a 20 cent line:

Chicago    +110
Cleveland  -130

The bookmaker bases the line on anticipated betting action. Suppose that $130 is bet on Cleveland for every $100 that is bet on Chicago; equivalently, 56.52% of the betting action is on Cleveland. If Cleveland wins, Chicago bets will pay off Cleveland bets and the bookmaker will break even; if Chicago wins, there will be $130 in losing Cleveland bets to pay for every $110 in winning bets, leaving the bookmaker with $20 for every $230 bet, or a profit equal to 8.7%. Thus, the bookmaker has a no-risk situation, provided that the betting action is properly divided.

How The Bookmaker Mitigates Risk

So how much slack is there in the betting action for the bookmaker to still be able to avoid risk? As we've seen, if 56.52% of the action comes in on the favorite, the bookmaker can't lose. But what if only 54% or 53% of the action comes in on the favorite?

It turns out that even if 52.38% of the betting action comes in on the favorite, the bookmaker will avoid risk: In this case, he will break even if the underdog wins and make a 7.3% profit if the favorite wins.

Do Bookmakers Get Greedy?

Maybe the bookmaker doesn't like breaking even and wants to always win. It can happen: The betting allocation that maximizes the bookmaker's guaranteed profit occurs when 54.27% of the betting action comes in on the favorite. In this case, the bookmaker will make a guaranteed profit of 4% no matter who wins the game. Unfortunately for the bookmaker, the bettors have a say in how much is bet on each team.

For the twenty cent line we've been discussing (-130, +110), the bookmaker can't lose if the percentage of betting action on the favorite is between 52.38% and 56.52%. When the action on the favorite is in this range, not only will the bookmaker avoid risk, he will make an average profit of about 4%.

If more than 56.52% of the action comes in on the favorite and the favorite wins, or if less than 52.38% of the betting action comes in on the favorite and the underdog wins, the bookmaker will lose. On the other hand, if the under bet team wins, the bookmaker will be a winner. In any event, whenever the percentage of action bet on the favorite strays outside the no-risk zone, the even the best sportsbooks are exposed.

Bottom line: The bookmaker can only avoid risk if the money line properly divides the betting action.