The formal study of probabilit
The formal study of probability began with questions regardinggambling and games of chance. The conventional analysis of gamblingis based on the expected values of these games which is alwaysnegative for the player and positive for the casino house. Theabsolute values of the two are exactly the same. Therefore, whatthe player loses equals what the house wins (in the long run). Ifthe expected value of a game for the player is 0, then the game is’fair’. Note that fair games would earn zero revenue for thecasino, so casinos cannot afford to provide players with fairgames! To earn revenue for the casino, games must be ‘unfair’, tothe advantage of the house. The ‘unfairness’ of casino games iswell-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 istypically played on a wheel with 38 slots numbered 00, 0, and 1through 36, although not in sequence. The 00 and 0 slots are green,and all other slots alternate in color, black/red/black (and soon), enabling players to place wagers many different ways. Thewheel is spun, then a ball is dropped onto the wheel and is equallylikely to end up in any one of the 38 slots.
There are many ways to bet and the payoffs are different fordifferent wagers. For example, to make a “straight” bet (payoff35:1), the chip(s) will be placed in one of the numbered spaces onthe game board, and if the ball ends up in that slot, the playerwins $35 for every $1 wagered. Note that the game of roulettereturns your initial bet to you if you win, so with this straightbet, a player who bets $1 will either have a gain of $35 or a lossof $1.
a) Marco decides to play roulette for the rest of the eveningand repeatedly places a $1 wager on the number 22. What is theexpected value of this game? (In other words, what is his expectednet gain over many, many repeated plays?) Explain why this is an’unfair’ game.
b) Maxine is a little less adventurous and hopes to win moreoften (and lose less often) so she repeatedly places her $1 bet onred (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 byplaying this way since the payoff is 1:1? Does she have a 50/50chance of winning each time the wheel is spun? What is the expectednet gain? Explain.
c) Recall last week’s discussion on The Law of Averages vs. TheLaw of Large Numbers and combine that with the questions that youjust answered. What do you think are some of the motivations behindgambling (that is, how do people justify gambling)?
Answer:
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 incolor, black/red/black (and so on), enabling players to placewagers many different ways.
a ball is dropped onto the wheel and is equally likely to end up inany 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 andrepeatedly places a $1 wager on the number 22
The probability that the ball lands on number 22 will be givenby,
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 thegame 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, $1won 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 isspun 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 sheshouldn’t expect to breakeven by playing this way.
c)
The gambler think that if they win one single round it willimmediately balance all the lost rounds, since the payoff inwinning round is more. But the law of large numbers tells us thatit is only valid for very large numbers.
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