Gaming

Gaming

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Recent years have indicated a change in the locality of legalized casino gambling across the United States. Until 1978 gaming, as it is commonly called, was available only in Nevada. Today, however, the casino business has expanded to other states. One reason for the increase in the number of gambling places is that many people enjoy playing these games of chance. Another critical factor is the profit margins realized by entrepreneurs in the casino industry.

Casinos offer a wide range of games to meet the interests of their clientele. Blackjack, roulette, baccarat, and craps are among the popular games. All casino games share two important characteristics that are critical to their

continued success. On the one hand, each game is exciting because the unpredictable nature of probabilistic outcomes makes it possible for a player to be lucky and win in the short run. On the other hand, the significant "house edge" associated with each of these games ensures that in the long run the bettor will lose and the casino will win money. To better understand the risks of casino gambling, consider the chances of winning in roulette and blackjack.

Roulette

The simplest of casino games to analyze from the perspective of theoretical probability is roulette. In this game a small ball is rolled onto a revolving tray containing thirty-eight numbered compartments. The ball randomly comes to rest in one of these thirty-eight compartments. In addition to the compartments numbered from 1 to 36, there are two extras numbered 0 and 00. It is these last two numbers that provide the casino with their edge.

The bettor has several options when playing roulette. If a particular number is chosen, the player will win $35 for every dollar bet if that number occurs. In this case, if a winning bet of $2 is placed the bettor will receive $72 in return, 35 to 1 on each dollar bet, plus the original $2 bet. The bettor may also choose to bet on the color of the number hit (eighteen are red and eighteen are black), whether the number is even or odd (0 and 00 are considered neither even nor odd), or whether the number hit will be 1 through 18, or 19 through 36.

In each of these bets, the probability of winning is 18/38 because in each case there are eighteen favorable results out of the total number of thirty-eight possible outcomes. Finally, the player can get 2 to 1 odds by betting that the number will be from 1 to 12, 13 to 24, or 25 to 36.

One interesting aspect of roulette is that the "strategy" used in choosing from the aforementioned betting options has absolutely no effect on the expected value of the win or loss on that bet. To show this, the expected value of each of these options can be calculated. In a nutshell, the expected value is the average amount that a player would win or lose on a bet. A positive expected value indicates that the game is favorable and that on average the player will win money, whereas a negative expected value indicates that the player will on average lose money.

If 0 and 00 were not included on the roulette wheel, the expected value on any wager would be 0, indicating that in the long run, the average win or loss for the bettor would be zero. To calculate the expected value, multiply the probability of each of the expected outcomes by the amount won or lost by the player given that outcome. Thus without 0 and 00, a player betting on even or odd would expect to win 50 percent of the time and lose the other 50 percent of the time.

If a bet of X dollars is made, the expected value is (0.5)(−X) + (0.5) (+X) = 0. Similarly, on the 36 number wheel, the expected win or loss on a bet of $X on an individual number is 0. The player wins of the time and loses of the time, making the expected value . Any game with an expected value of 0 is referred to as a fair game and will not be found in any casino!

The green numbers 0 and 00 provide the casino with its edge. If a bet is made on an individual number, the probability of winning is , since there are now 38 equally likely outcomes that can occur. Calculating the expected value for betting $X on an individual number yields the following equation: . Thus the expected value is approximately 0.053 times the amount of the wager. This means that the player on average will lose a little more than 5 cents on every dollar bet.

An even or odd bet has a probability of of winning since 0 and 00 are considered neither even nor odd. Calculating the expected value for the player betting $X on even or odd yields the following equation: . Thus the expected value on this type of bet is exactly the same as that of betting on an individual number. The reader is encouraged to perform the appropriate calculations to verify that betting on 1 through 12 also yields an expected value of −0.053 multiplied by the amount of the bet. In roulette, therefore, no matter what strategy is used, the disadvantage to the player remains constant.

Some gamblers, however, feel they can win by trying to determine what is "due" to happen on a particular trial. For example, if the previous four numbers have been red, some feel that black is more likely to occur the next time. They reason that because attaining red five times in a row is extremely unlikely, the next result is more likely to be black. The fallacy here is that although the probability of red occurring five times in a row is indeed very unlikely, the unlikely event of four consecutive reds has already occurred.

Because successive trials of a roulette wheel are independent of one another, the next number has the same chance of being red or black as in previous trials, . Thus playing this "hunch" strategy does not make the player any more (or less) likely to win as the expected value for each trial remains −0.053 times the amount bet.

Blackjack

Blackjack is perhaps the most complex of casino games and for this reason has been the subject of considerable analysis by gamblers. Calculating the expected value of a hand of blackjack is an extremely difficult task for several reasons. First, there are many ways in which a hand can be dealt to the player and the dealer. Second, almost every casino offers slightly different rule variations of the game. Finally, unlike roulette, the decisions made by the player throughout the course of a hand dramatically affect the chances of winning.

In the game of blackjack, picture cards have a value of 10, aces can be counted as 1 or 11, and all other cards take on the value of the card. The game begins with the player and the dealer each being dealt two cards. The player's goal is to come closer to 21 than the dealer without going over, known as busting. After seeing both of their cards and one of the dealer's cards (commonly called the "dealer's up card"), the player must decide how to proceed. The player can "stay" with the present total, take another card ("hit"), double the wager and be given one and only one more card, or split a pair by placing another bet equal to the original and then proceeding with two separate hands.

In the majority of cases, the only reasonable decision is to take a hit or stand with the current total. Should the player take a hit, he or she can take additional hits until either the total of the cards exceeds 21 or the player does not want any more cards. The player loses immediately if the sum of the cards exceeds 21. If the player stops taking cards without exceeding 21, it is the dealer's turn to act. In casino versions of blackjack, the dealer does not make decisions but instead the dealer's play is fixed. The dealer must hit whenever her total is 16 or less and must stay when her total is 17 or more.

Many books on blackjack provide a basic strategy that is considered optimal for the player. Optimal basic strategy is determined by considering every possible combination of the player's hand and the dealer's up card and making the play that yields the best expected value in each case. Depending on the rule variations, most authors claim that by playing a sound basic strategy the house advantage will only be about 2 to 3 percent. Although this disadvantage is less than that of roulette, we must keep in mind that it is based on the player making the correct decisions at all times. Unlike roulette, the quality of the decisions made in the course of playing has a dramatic effect on the chances of winning. In practice, most players do not consistently make the optimal decisions and for this reason, many blackjack players lose at a much faster rate than they otherwise should.

How the House Stays Ahead

A simple example will show how the casino gains its advantage as the number of games played increases. Suppose we consider a situation in which the player has a 40 percent chance of winning and a 60 percent chance of losing each game with the amount won or lost in each game being equal. If three games are played, the player must win two or three games in order to be ahead. The probability of this occurring is 3(.4)(.4)(.6) + 1(.4)(.4)(.4) = 35 percent, because there are three ways in which the player can win two out of three games. That is, the probability of this occurring is (.4)(.4)(.6) for each of the three ways, and there is one way of winning all three games, the probability of which is (.4)(.4)(.4). Similarly, the probability of winning three or more games in five is

10(.4)(.4)(.4)(.6)(.6) + 5(.4)(.4)(.4)(.4)(.6) + 1(.4)(.4)(.4)(.4)(.4) = 32 percent.

Further analysis of more games played can be done using the binomial distribution . The successive probabilities of the player winning when the game is played 7, 9, 11, 13, 15, 17 and 19 times are 0.29, 0.266, 0.246, 0.228,0.212, 0.198, and 0.187, respectively.

The probability of winning decreases with the duration that a gambler plays. In a casino, a gambler may play hundreds of hands of blackjack or games of roulette consecutively, yet the chances of the player being ahead after a lengthy session are extremely small. Furthermore, from the casino's perspective, thousands of hands of blackjack and games of roulette are being played every hour by their numerous patrons, nearly around the clock. Given this huge number of trials, the casinos are assured of making large profits.

see also Probability and the Law of Large Numbers; Probability, Theoretical.

Robert J. Quinn

Bibliography

Khazanie, Ramakant. Elementary Statistics: In a World of Applications. Glenview, IL: Scott, Foresman and Company, 1979.