Figuring out Odds (Meeting 2, 4/6/05) J. Pasquale - Casinos advertise paying odds for possible bets - This is what the casino pays for your winning bet - 3 to 1: casino pays $3 to your $1 bet - Some examples - Pass line bet in craps pays 1 to 1 - Casino pays $1 if you bet $1 and win - Winning single number in roulette pays 35 to 1 - Casino pays $35 if you bet $1 and win - Winning hand in blackjack pays 3 to 2 - Casino pays $3 if you bet $2 and win - The odds the casino offers are (almost always) NOT the same as "fair odds", i.e., what would be paid in a "fair game" - How do we determine fair odds? - Say P is probability of winning - You place a $1 bet - If you lose, you lose your $1 - If you win, you win X (and you get your $1 back) - Calculating expected gain, EG - EG = PX - (1 - P) = PX - 1 + P = P(X + 1) - 1 - This is probability of winning times X - Minus probabilty of losing times $1 - If EG > 0, this is a "positive expectation game" - Over long run, you will make money - If EG < 0, this is a "negative expectation game" - Over long run, you will lose money - All casino games are negative expectation games - If EG = 0, this is a "fair game" - Over long run, your gain is zero - Neither side has an advantage (or disadvantage) - Determing fair odds - Start with P (X + 1) - 1 = 0 - A bit of algebra - P (X + 1) = 1 - PX + P = 1 - PX = 1 - P - Result: X = (1 - P) / P - Examples - If P = 1/3, X = (1 - 1/3) / (1/3) = 2/3 * 3/1 = 2 - This means you must win $2 for every $1 bet - Fair odds are 2 to 1 - If P = 1/2, X = (1 - 1/2) / (1/2) = 1/2 * 2/1 = 1 - This means you must win $1 for every $1 bet - Fair odds are 1 to 1 - If P = 2/3, X = (1 - 2/3) / (2/3) = 1/3 * 3/2 = 1/2 - This means you must win $0.50 for every $1 bet - Fair odds are 1/2 to 1, more conveniently expressed (using whole numbers) as 1 to 2 - If P = 5/6, X = (1 - 5/6) / (5/6) = 1/6 * 6/5 = 1/5 - This means you must win $0.20 for every $1 bet - Fair odds are 1/5 to 1, or 1 to 5 - More generally - Given P, fair odds are (1 - P) / P to 1 - If P = 1/N, X = (1 - 1/N) / (1/N) = (N-1)/N * N/1 = N-1 - Fair odds are N-1 to 1 - If P = (N-1)/N, X = (1 - (N-1)/N) / ((N-1)/N) = 1/N * N/(N-1) = 1/(N-1) - Fair odds are 1 to N-1 (derived from 1/(N-1) to 1) - Some Practice - P Fair Odds - 1/2 1 to 1 - 1/3 2 to 1 - 1/4 3 to 1 - 1/5 4 to 1 - 1/6 5 to 1 - 2/3 1 to 2 - 3/4 1 to 3 - 4/5 1 to 4 - 5/6 1 to 5 - But, casinos do not pay based on fair odds! - Say casino gives 1 to 1 odds, but fair odds are 2 to 1 - P = 1/3 (your probability of winning) - EG = (1/3)1 - (2/3)1 = -1/3 - "House edge" is 33% - Over the long run, you will lose 33% of your money - Say casino give A to B odds, but fair odds are C to D - if B/(A+B) > D/(C+D), house has the edge, and you will lose over the long run - Typical casino bets and house edges - Roulette: bet on "red" pays 1 to 1 - But P = 47.4% (18 out of 38) - Fair odds are (1-P)/P to 1, or 1.11 to 1, i.e., they are supposed to be paying you 1.11 to 1, and not 1 to 1! - House edge - EG = (18/38) 1 - (20/38) 1 = .053 or 5.3% - In the long run, you will lose 5.3 cents for every dollar bet - Craps pass line bet: 1.4% - Craps pass line bet with 2x odds: 0.6% - Craps Proposition 7 bet: 16.7% - Blackjack: 0.2% - Baccarat Player: 1.24% - Keno: 25-29% - Roulette: 5.3% - Video poker: 0.5% (assuming you trust the machine!) - Slot machines: 2-15% - House edge doesn't tell the whole story! - Say house edge is 0, i.e., odds = fair odds - Say fair odds are 99 to 1 (house edge is 0) - P = 1/100 - What this means is that usually, with prob 99/100, you will lose - But with prob 1/100, you will win, and when you do, you win big - Bet $1 to win net of $99 - Say fair odds are 1 to 99 (house edge is 0) - P = 99/100 - What this means is that usually, with prob 99/100, you will win - So, with prob 99/100, you will win, but when you do, you win smal - Bet $99 to win net of $1