Adam Bowers | Department of Mathematics | UC San Diego

Basic Rules for Casino Games

These rules are provided to help you solve probability problems related to simple games of chance that appear in the various textbooks we use for probability and statistics courses

Roulette   (wikipedia article)

A roulette wheel is marked with the numbers 1, 2, 3, ..., 36. Half of the numbers are in black and the other half of the numbers are in red. The idea is that you can either bet on a specific number or you can bet on a color. For example, if you bet on red and the wheel lands on red, you win 1 dollar; however, if the wheel does not land on red, then you lose 1 dollar.

If the wheel had only these 36 numbers, then the game would be fair. Usually, however, casino games are not fair. The house always has a slight advantage (in this case very slight). A roulette wheel will have either one or two green numbers. These are the numbers 0 and 00.

American casinos (like those in Las Vegas) use both 0 and 00, so the probability of landing on red (and winning the bet) is 18/38. The probability of landing on a color that is not red (i.e., on black or green) and losing the bet is 20/38.

In French casinos (such as those in Monte Carlo) they use only the 0. This means that the probability of winning a bet on red is 18/37 and the probability of losing is 19/37. Las Vegas casinos have a slightly larger house advantage than the Monte Carlo casinos.

Dice   (wikipedia article)

Usually, the dice game at casinos is "Table Craps" (or just "Craps"). In the game of Craps, the player (or "shooter") rolls two fair six-sided dice and computes the sum of the two numbers showing on the dice. If the player rolls a sum of 7 or 11 (known as a "natural"), the player wins. If the player rolls a sum of 2, 3, or 12 (known as "rolling craps"), the player loses. Any other roll (a sum of 4, 5, 6, 8, 9, or 10) is a "point number". If the player rolls a point number, the objective of the game changes, and the player tries to roll the same point number again before rolling a sum of 7.

Each die can land on any number in the set {1, 2, 3, 4, 5, 6}. This means there are 36 possible outcomes, usually written in the form (x,y), where x is the number on the first die and y is the number on the second die. All 36 outcomes are equally likely.

When playing the game, we are not interested in the outcome of the roll (which is a pair (x,y)) so much as we are interested in the sum (S=x+y). On the opening roll, we win if S=7 or S=11 and we lose if S=2, S=3, or S=12. If S is any other number, then the new objective is to achieve the same sum again. So, if S=4 on the opening roll, the objective is to roll S=4 again. This can be done with (3,1), (2,2), or (1,3). Any of these three outcomes works. The player does not need to obtain the same outcome as before (just the same sum).

The table below gives the probabilities for each possible value of S:


Poker   (wikipedia article)

There are many, many different kinds of poker, so I will only discuss the terminology. I will not discuss the details of any particular poker game. (See here for a list of poker variants.)

A standard poker deck contains 52 cards. Each card has a "suit" and a "rank". There are four suits and 13 ranks. The four suits are {♥, ♦, ♣, ♠}. (These are called Hearts, Diamonds, Clubs, and Spades, respectively.) The 13 ranks are (in order from lowest to highest) {2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A}. (J is Jack, Q is Queen, K is King, and A is Ace.) Each suit and rank pair leads to one card, so A♠ denotes the Ace of Spades.

A "poker hand" consists of a combination of five cards taken from the deck (usually at random). Certain poker hands have names. A few examples are:

  1. When we say that five cards have sequential rank, we mean that the ranks are consecutive. For example, 2-3-4-5-6 and 8-9-10-J-Q are consecutive, so these have sequential rank. An Ace can count as either a high card (following the King) or a low card (preceding a 2), but it cannot function as both at the same time. Thus, A-2-3-4-5 and 10-J-Q-K-A are admissible sequences for a straight or a straight flush, but Q-K-A-2-3 is not.
  2. A straight flush is neither a straight nor a flush. So, if someone asks for the probability that five randomly chosen cards form a straight, you must exclude the possibility that the five cards all have the same suit. Similarly, if you want to compute the probability that five randomly selected cards form a flush, you must exclude the possibility that the cards are in sequence.
  3. A straight flush where the ranks are 10-J-Q-K-A is known as a royal flush. If someone asks you to compute the probability of a random poker hand being a straight flush, you should ask them if they want you to include a royal flush, which is sometimes considered separate.
(See here for a complete list of poker hands.)