Roulette and baseball are popular wagering activities, yet, mathematically, they are quite different.
An American roulette wheel has 38 sections labeled 1 through 36, 0 and 00. Suppose you bet on "17." Since there is only one section marked 17, your chance of winning equals 1 in 38 and your chance of losing equals 37 in 38.
Since your chance of winning equals 1/38, true odds are 37 to 1. If payoff odds equaled true odds, then, on the average, winning bets would be paid by losing bets and the casino would break even. In order to insure an edge, casino payoff odds on single number roulette bets are 35 to 1 rather than 37 to 1.
In repeated $1 bets on "17," a gambler will win about once in 38 bets ($35 payoff) and lose about 37 times in 38 bets ($37 loss) for an average loss of $2 per 38 bets. This number, expressed as percent profit, is called the house edge. For this bet, house edge = 2/38 = 5.3%. If you bet repeatedly, then in the long run, you will lose about 5.3 cents per dollar bet. You may get lucky and be a short term winner, but in the long run, you will lose.
Baseball is different. Unlike roulette, there is no mathematical way to determine the probability that you will win a bet. In fact, your chance of winning varies from team to team, pitcher to pitcher, and game to game, so the bookmaker can't rely on the laws of chance to insure a long run profit.
In baseball betting, payoff odds are expressed as money lines. For example:
San Francisco -130 means that if you bet on the favorite, San Francisco, you must bet $130 to win $100. Florida +130 means that a winning $100 bet on the underdog, Florida, yields a $130 profit.
If the money line for the favorite is the negative of the money line for the underdog, you get the sports version of true odds: If appropriate amounts are bet on each team in a game, losers pay winners and the bookmaker breaks even. For example, using the money lines SF -130, Florida +130, if 10 gamblers each bet $130 on San Francisco and 10 gamblers each bet $100 on Florida, losing wagers pay winning wagers, leaving no profit for the bookmaker.
Analogous to roulette, in which casino payoff odds are slightly lower than true odds, for baseball bets the bookmaker lowers the money line on the underdog:
San Francisco -130
With the new line, if 10 bettors each bet $130 on San Francisco and 10 bettors each bet $100 on Florida, if Florida wins, the $1,300 in losing bets on San Francisco pay the $1,100 in winning bets on Florida, leaving the bookmaker with a profit of $200 out of $2,300 bet = 8.7%. If San Francisco wins, the $1,000 in losing Florida bets pays winning bets and the bookmaker breaks even.
Thus, if $1,300 is bet on San Francisco for every $1,000 bet on Florida, the bookmaker can't lose and will make a profit of 8.7% if Florida wins.
Assume that the money line for SF reflects the true probability that San Francisco will win. Then SF -130 means that San Francisco should win about 13 times in 23 and lose 10. Since the bookmaker will make a profit of 8.7% whenever SF loses and break even whenever SF wins, the bookmaker's average profit for this bet equals 8.7%x(10/23) + 0x(13/23) = 3.8%.
In roulette, the casino makes a long run profit of 5.3% on single number bets, no matter what, whereas in the baseball bet we've been discussing, the 3.8% average profit assumes that the betting action is properly divided (It is thus important for the bookmaker to set lines that properly divide the action). Because of this, the bookmaker's average profit is usually called the "hold percentage" rather than the house edge.
The 3.8% hold percentage that we calculated here depends not only on the betting allocation, but also on the money line itself. For higher money lines (bigger favorite), the bookmaker's profit is a smaller fraction of the average take and so the hold percentage is lower. For lower money lines (more evenly matched games) the hold percentage is higher.
Where does the money line come from in the first place? There is no mathematical formula by which the bookmaker can calculate true odds (You could even argue that in sports true odds don't exist). Instead, the bookmaker uses his knowledge of baseball and betting behaviors to come up with the correct number to appropriately divide the betting action.
In roulette, it doesn't matter who bets on what for the casino to make a long run profit. In baseball, if money lines don't appropriately divide betting action and the bettors are smart, the bookmaker will lose. And even if the betting public is foolish, since money lines are set only to divide the betting action, a smart bettor can make a profit at the expense of naive bettors even when the bookmaker makes a profit.