Question

The formal study of probability began with questions regarding gambling and games of chance. The conventional analysis of gambling is based on the expected values of these games which is always negative for the player and positive for the casino house. The absolute values of the two are exactly the same. Therefore, what the player loses equals what the house wins (in the long run). If the expected value of a game for the player is 0, then the game is 'fair'. Note that fair games would earn zero revenue for the casino, so casinos cannot afford to provide players with fair games! To earn revenue for the casino, games must be 'unfair', to the advantage of the house. The 'unfairness' of casino games is well-known to players. The players, however, knowingly play the 'unfair' games!

(Reference: http://www.casinosprofit.com/the-expected-value-of.html )

Consider the game of roulette, a well-known casino game. Originating in late seventeenth-century France, this game is typically played on a wheel with 38 slots numbered 00, 0, and 1 through 36, although not in sequence. The 00 and 0 slots are green, and all other slots alternate in color, black/red/black (and so on), enabling players to place wagers many different ways. The wheel is spun, then a ball is dropped onto the wheel and is equally likely to end up in any one of the 38 slots.

There are many ways to bet and the payoffs are different for different wagers. For example, to make a "straight" bet (payoff 35:1), the chip(s) will be placed in one of the numbered spaces on the game board, and if the ball ends up in that slot, the player wins $35 for every $1 wagered. Note that the game of roulette returns your initial bet to you if you win, so with this straight bet, a player who bets $1 will either have a gain of $35 or a loss of $1.

a) Marco decides to play roulette for the rest of the evening and repeatedly places a $1 wager on the number 22. What is the expected value of this game? (In other words, what is his expected net gain over many, many repeated plays?) Explain why this is an 'unfair' game.

b) Maxine is a little less adventurous and hopes to win more often (and lose less often) so she repeatedly places her $1 bet on red (which has more ways to win but a winning payoff of only 1:1, $1 won for every $1 bet). Should she expect to break even by playing this way since the payoff is 1:1? Does she have a 50/50 chance of winning each time the wheel is spun? What is the expected net gain? Explain.

c) Recall last week's discussion on The Law of Averages vs. The Law of Large Numbers and combine that with the questions that you just answered. What do you think are some of the motivations behind gambling (that is, how do people justify gambling)?

Answer 1

Given data

The expected value of a game for the player is 0, then the game is 'fair'.

The game is typically played on a wheel with 38 slots numbered 00, 0, and 1 through 36, although not in sequence.

The 00 and 0 slots are green, and all other slots alternate in color, black/red/black (and so on), enabling players to place wagers many different ways.

a ball is dropped onto the wheel and is equally likely to end up in any one of the 38 slots.

a player who bets $1 will either have a gain of $35 or a loss of $1.

a)

Marco decides to play roulette for the rest of the evening and repeatedly places a $1 wager on the number 22

The probability that the ball lands on number 22 will be given by,

The expected gain in this case would be $35.

Hence the expected value will be given by,

Here we can see that the expected value is negative and hence the game is unfair.

b)

Maxine is a little less adventurous and hopes to win more often (and lose less often) so she repeatedly places her $1 bet on red (which has more ways to win but a winning payoff of only 1:1, $1 won for every $1 bet).

The probability of getting a red will be given by,

She doesn't have a 50-50 chance of winning each time the wheel is spun since the above probability is less than 0.50.

The expected net gain will be given by,

Hence for Maxine too the net gain is negative. Therefore she shouldn't expect to breakeven by playing this way.

c)

The gambler think that if they win one single round it will immediately balance all the lost rounds, since the payoff in winning round is more. But the law of large numbers tells us that it is only valid for very large numbers.