Make your own free website on

The Math Learning Portfolio Casino: Test Your Luck

Probability Laws
Four Probability Properties
Pascal's Wager and Bernoulli Trials
John Forbes Nash and Game Theory
Chance Games
Math in Music
Demonstration of Goals

Chance in the Casino

Chance Games

Gamblers playing Roulette

Here is a typical slot machine that you would find in a casino.



Roulette is played by making bets on likelihood of where a ball will land when tossed onto a wheel. This mounted horizontal wheel consists of 38 evenly divided, marked from 0 to 36 and one additional section marked 00. Sections 1 to 36 are coloured red or black in an alternating pattern, while 0 and 00 are coloured green. There is also a betting layout beside the wheel for the players to place their betting chips.


To find out how much money a player has the chance of winning, one has to consider the payoff odds and the probabilities involved. Payoff odds are usually set lower than the actual probability of winning by a small amount so that casinos have a long-term profit, but gamblers still have a fairly good chance as well. For example the true chance of landing a red would be 20 to 18 because there are 18 red sections and 20 non-red sections, but the payoff odds could be 1 to 1, a slightly lower value. Another example would be bets on 19 where there is one section marked 19 and 37 non-19 sections, meaning a 37 to 1 chance, but the payoff odds could be 35 to 1. Let’s expand this idea further:


Bets on Red

Bets on 19

Suppose a gambler bet $1.

Since the payoff odds are 1 to 1, the gambler would have an average loss of $2, which translates to -5.3 cents/dollar bet (2/38), the expected pay off. Conversely 5.3 cents/dollar bet is the casino’s expected pay off, also called house edge.

Suppose a gambler bet $1.

Since the payoff odds are 35 to 1, the gambler would have an average loss of $2, or of 5.3 cents/dollar bet. However, the casino will receive just the opposite, a house edge of 5.3 cents/dollar bet.

Even though, the probabilities of each of these results are different, casinos can maintain the same house edge no matter. Therefore, in terms of expected payoff, these bets are the same.


Another way to find the expected payoff:

$1 Bet on Red

Win                     Probability

1                          18/38

-1                           20/38

Expected payoff= (1 x 18/38)    (1 x 20/38) = -2/38 = -.053


            Players have used many strategies to try and win in roulette. Some of them include using simultaneous bets, doubling up, and even using a computer. People who cast simultaneous bets hope that by betting on many sections of the wheel in one spin, they have a greater chance of winning. However, as averages are additive, your average losses will also grow. By doubling up, the player would first bet $1. If the player loses, then bet double the amount and continue until a win, to cover the losses and retain a profit of $1. This strategy does not work however if there is a maximum number of bets that one can place. Some of the more ambitious gamblers might try using computers to calculate the trajectory at which the ball hits the wheel to then give a good prediction of where it will land. However, it would require that bets be placed after the ball is thrown, not to mention that casinos do not tolerate such behaviour. Therefore, there are no real strategies for winning Roulette.


Slot Machines


            Slot machines are either mechanical or computerized programs with three or more reels of possible outcomes such as sevens, cherries, and dollar signs. Once a player inserts a coin and pulls the lever to the side or pushes a button, the reels begin to spin quickly and once they stop, whatever configuration you have, determines your wins or losses. Usually there is a list of winning configurations at the front of the machine, and if you are one of the lucky winners, you receive a payoff.


            There are no real strategies to winning slot machines as they are quite random and can often be altered by the flick of a switch or some reprogramming. However, one might consider using the Monte Carlo Method to find the house edge of certain machines to increase the chance of winning. The Monte Carlo Method is repeating an activity many times over to estimate the likeliness of occurrence, because as the law of averages points out, the more repetitions, the closer one will get to the expected outcome. For example:


Playing a $0.25 Slot Machine

Amount of $ Used = $50 Number of Plays Required = 430

House Edge in = 50/ 430/ 25 x 100 = 46.51 cents/ dollar bet


Video Poker


            Video poker is usually draw poker or other type simulated on the computer. The game starts by inserting a coin and making a bet. Then the computer, using a random-number generator, gives the player five cards. The player can then decide to keep or discard certain cards by pressing a button or by using a touch screen. The player wins when they hold a card configuration on the payoff table, located usually on the front of the machine. Video poker is different from regular poker in that there is no interaction with another human so players cannot use techniques like bluffing. In video poker, bets are placed before game play and no folding is allowed. In addition, the casino pays players, and usually takes a percentage of each pot.


            The only strategy one can employ in video poker is deciding which cards to hold or discard. The aim of the strategy is to give the smallest possible house edge while the aim of the machine is to yield the highest possible house edge. To accomplish this, one has to find the probability of getting the desired configurations of cards when delivered every possible initial hand. Then you would compare the probability to the payoff odds (and see which has the lowest house edge) to find which configuration you would want to aim for.


            Another form of video poker is the progressive game where a network of video poker machines (controlled by a central computer) plays for a jackpot (usually a royal flush), where a few cents of each bet is put toward it as long as no one wins. Jackpots can often reach millions because the chance of winning is incredibly low. These progressive games are the casino’s way of competing with lotteries. However, they are slightly different in that there are no multiple winners; whoever hits the jackpot first, wins the entire amount – a positive expected payoff. A drawback of the game is that the larger the number of people on the network, the smaller the expected payoff and this is where team play can be of use.




            Blackjack is based on the number 21 and is played between one or more gamblers and the dealer. Players try to have a higher score than the dealer without going higher than 21, the ideal score. 2 to 10 are worth face value; jacks, queens, and kings are worth 10, aces are either worth 1 or 11, and to calculate scores, you find the sum of the cards’ values. Gamblers make bets by placing chips onto the circle section of the betting layout. Then the dealer gives out two cards to each player and deals two cards to him/herself, one face up (“up card”) and one face down (“hole card”). The first player to the left of the dealer chooses whether to “stand” (take no additional cards) versus “draw” (take another card). Then the game continues on, but if you ever “bust” (get a score higher than 21), then you lose. If no one has busted, then the dealer draws. The dealer must draw with a score under 17 and stand with a score of 17 or up. Then if the dealer busts, anyone who has not wins. If the dealer does not bust, anyone above it wins, while anyone below it loses. Players with the same score tie, and chips remain with their respective players. In the case of aces, if you play the ace as worth 11, then it is called “soft”, while aces worth 1 are called “hard”. You can double down after being dealt the first two cards, meaning that by doubling your bet, you will get one face down card that you will compare with the dealer after he/she plays. If you get two cards of the same type (e.g. two 8s), then you may split your pairs, meaning that you now have two separate hands, dealt face up from now on. If the first two cards are an ace and a card with a value of 10, then you have a “natural” or “blackjack” hand; if you have a natural hand and the dealer does not, then you win. If the dealer has an ace as his/her up card, then the players can make insurance bets that the hole card is a 10-value card by placing half your chips in that section of the betting layout. In this way, you can protect yourself from losing if you do not have a natural and the dealer does. However, as with all bets, there is a possibility of loss, so you should not make an insurance bet unless you know that the deck is particularly rich of 10-value cards at that moment.


            With Edward O Thorp’s Beat the Dealer, gamblers received winning strategies to playing blackjack. These winning strategies were developed because the dealer has more restrictions than the player (the dealer must draw with a score of 16 or less, while players do not have to). So when the dealer has to bust, the player can draw fewer times to ensure that they will not bust. Players can also increase their chance of winning by counting the cards as they are played, so that they know what the deck is comprised of and then whether to stand or to draw.



Here is the typical strategy provided in blackjack books:


Drawing and Standing

1.       Draw with a hard hand of 11 or less.

2.       Draw if you have a hard hand with a score between 12 and 16 and the dealer’s up card is 7 or higher, including ace.

3.       Draw if you have a soft hand of 17 or less.

4.       Draw if you have a soft 18 and the dealer’s up card is a 9, 10, or ace.

5.       Stand if none of these conditions are met.


Doubling Down: Takes Precedence over Drawing.

1.       Double down if your score is 11.

2.       Double down if your score is 10 and the dealer’s up card is 9 or less.

3.       Double down if your score is 9 and the dealer’s up card is 3 through 6.

4.       Double down if your fist two cards are an ace and a 2 through 7 and the dealer’s up card is a 4,5, or 6.


Splitting Pairs: Splitting Pairs Takes Precedence over Drawing.

1.       Always split a pair of aces or 8s.

2.       Never split a pair of 4s, 5s, or 10s.

3.       Split a pair of 2s, 3s, 6s, or 7s when the dealer’s up card is 3 through 7

4.       Split a pair of 9s when the dealer’s up card is 2 through 6, 8, or 9.



Do not buy unless fraction of 10-value cards in deck is less than 1/3. Do not attempt buying insurance if you do not count cards


From: Mike Orkin’s What Are the Odds?, p. 91




A web version of the game blackjack

math is fun!