Thursday, July 22, 2010

Wednesday Math (one day late), Vol. 122: How bookies make money

Bookies are sometimes called oddsmakers, but they don't really make probabilities, they make payoffs for events. If I am going to have a "friendly wager" with someone, the two of us will agree on some event we and decide how much money each of us is going to have to risk. Often, we both put in the same amount of money, which back in the days when horse racing was one of America's most popular sports would be called a 1-1 (or 1:1) wager. The first number is the profit and the second number is the amount risked. We could also have a wager where one of us puts in more money because we think the event the other person is betting on is unlikely. For example, if you win only if you roll a 6 on a six-sided die, I expect to win five times out of every six and you expect to win only one time in every six. To make this "fair", we should make this a game where I risk 5 units to win 1, and you risk 1 unit to win 5. From my point of view, this is a 1:5 wager, while from your point of view, it's a 5:1 wager.

Betting with a bookie is not a friendly wager. Bookies take action from either direction and instead of all the money in the till going to the winner, the bookie hopes to take a little slice regardless of the outcome. This can only happen if they have a certain percentage of money on each side so they can cover the winners' payoffs and still have a little extra to pay the rent.

Many wagers are no longer defined in the classic profit:risk still seen when looking at the Racing Form. Instead, a modern gambling instruction line will look like

SEATTLE -140______Chicago + 130

Translating into English. Seattle is at home (all caps) and they are the favorite to win. -140 means any bettor favoring Seattle has to risk \$140 to profit \$100. +130 means Chicago bettors will make a profit of \$130 from a \$100 bet. The pot is sweetened because Chicago is likely to lose this game, according to some popular opinion. If there is only one bettor of the same level on each side, here is the situation.

Amount in the pot: \$140 + \$100 = \$240
If Seattle wins: Seattle bettor gets all \$240
If Chicago wins: Chicago bettor gets \$230, bookie pockets \$10

So if the money stays at this level on both sides, it's like the bookie is hoping for Chicago to win. If it's ten bets on each side, the bookie can't lose money, but can only win \$100 if the underdog Chicago team wins.

Here's a different situation. If it was 13 bettors on Seattle and 14 bettors on Chicago, the total pot would be 13*\$140 + 14*\$100 = \$3,220. If Chicago wins, the 14 winners get \$230 each, and that takes all the money out of the pot. If Seattle wins 13*\$240 = \$3,120, and the bookie pockets \$100 in case of a Seattle win.

If we add these two situations together, 23 Seattle bettors and 24 Chicago bettors is the ideal situation, since no matter who wins, the bookie will get \$100. This is the sweet spot for the oddsmaker, and the only "bad thing" that can happen is the game gets rained out, because then he has to give all the bets back.

Hey, who's feeling lucky?