Chapter 6
Craps
Craps is the most popular dice game in casinos. The game has been featured in multiple movies including Ocean's Thirteen (2007), Snake Eyes (1998), and Big Town (1987). The mathematical analysis of the game of craps is similar in some ways to that of roulette (both games involve independent rounds of play), but because each round is composed of 2 interdependent phases that have different rules, the analysis has to be carried out carefully.
6.1 Rules and Bets
In craps you are betting on the outcome of two dice rolled simultaneously. An appealing feature of the game is that you can play it either as the shooter (if you are the one rolling the dice) or as a stand-by (if you are a spectator, by betting with or against the shooter). As with roulette, players place their bets by placing their chips on the appropriate sections of the board (see Figure 6.1). The nicknames associated with each of the outcomes are presented in Table 6.1.
Table 6.1 Names associated with different combinations of dice in craps
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | Snake eyes | – | – | – | – | – |
| 2 | Ace ddeuce | Hard four | – | – | – | – |
| 3 | Easy four | Five (fever five) | Hard six | – | – | – |
| 4 | Five (fever five) | Easy six | Natural or seven out | Hard eight | – | – |
| 5 | Easy six | Natural or seven out | Easy eight | Nine (nina) | Hard ten | – |
| 6 | Natural or seven out | Easy eight | Nine (nina) | Easy ten | Yo (Yo-leven) | Boxcars or Midnight |
Figure 6.1 The layout of a craps table.
Each round of craps is comprised of two phases. The first phase consists of a single roll called the come-out roll, and the second phase (which might consist of multiple rolls) is called the point.
6.1.1 The Pass Line Bet
The pass line (also called win or right) bet is the most basic bet in craps, and the shooter is obligated to make this wager in order to play. In addition to the shooter, any spectator can participate in the pass line bet. Typically, the pass line bet pays even odds (recall that this means that, if you win, you get your money back plus a profit equal to your bet).
The outcome of the pass line bet is resolved as follows. If the come-out roll is a 7 or 11 (called a natural), then the pass line bet wins automatically, and the round ends (there is no second phase in that case). Similarly, if the come-out roll is a 2, 3, or 12, then the pass line bet loses automatically, and again the round ends. Losing in this way is often referred to as crapping out. Finally, if any other number is rolled (i.e., a 4, 5, 6, 8, 9, or 10), that number becomes the point and we move into the second phase of the round.
When a point is established, the goal of the game changes. The shooter keeps rolling the dice until either the point comes up again or a 7 comes out. If the point comes out first, then the pass line bets win. On the other hand, if a 7 comes out first (referred to as seven out), then the pass line bets lose. Note that this is the opposite of what happens in the come-out roll, where a 7 wins the game.
Let's analyze the pass line bet and compute the house advantage in craps. First, recall that there are 36 outcomes for the roll of two dice, and that, as long as the dice are fair, they are all equiprobable (see Table 6.2). To analyze craps, it is convenient to group these 36 outcomes into 11 groups, depending on what their sum is (see Table 6.3).
Table 6.2 All possible equiprobable outcomes associated with two dice being rolled
| 1–1 | 2–1 | 3–1 | 4–1 | 5–1 | 6–1 |
| 1–2 | 2–2 | 3–2 | 4–2 | 5–2 | 6–2 |
| 1–3 | 2–3 | 3–3 | 4–3 | 5–3 | 6–3 |
| 1–4 | 2–4 | 3–4 | 4–4 | 5–4 | 6–4 |
| 1–5 | 2–5 | 3–5 | 4–5 | 5–5 | 6–5 |
| 1–6 | 2–6 | 3–6 | 4–6 | 5–6 | 6–6 |
Table 6.3 Sum of points associated with the roll of two dice
| Sum of the dice | List of ways the sum can be realized | Number of ways the sum can be realized |
| 2 | 1–1 | 1 |
| 3 | 1–2, 2–1 | 2 |
| 4 | 1–3, 3–1, 2–2 | 3 |
| 5 | 1–4, 4–1, 2–3, 3–2 | 4 |
| 6 | 1–5, 5–1, 2–4, 4–2, 3–3 | 5 |
| 7 | 1–6, 6–1, 2–5, 5–2, 3–4, 4–3 | 6 |
| 8 | 2–6, 6–2, 3–5, 5–3, 4–4 | 5 |
| 9 | 3–6, 6–3, 4–5, 5–4 | 4 |
| 10 | 4–6, 6–4, 5–5 | 3 |
| 11 | 5–6, 6–5 | 2 |
| 12 | 6–6 | 1 |
As we did for the Monty Hall problem, we can use a tree to help us compute the probability of winning at craps (see Figure 6.2). We proceed now to fill in the probabilities associated with each of the branches in the tree.
Figure 6.2 Tree representation for the possible results of the game of craps. Outcomes that lead to the pass line bet winning are marked with W, while those that lead to a lose are marked L.
Since all 36 outcomes are equiprobable, it is easy to see that the probability of getting a natural in the come-out roll (i.e., winning in the first phase if you make the pass line bet) is simply equal to
Similarly, we can compute the probability of crapping out
and the probability of getting each of the points is
Filling these numbers into the tree we obtain Figure 6.3. Note that these probabilities sum to 1, as we would have expected. Also, note that the probability of winning during the come-out roll (
) is much larger than the probability of losing during the come-out roll (
), and that both are much smaller than the probability that a point will be made and we move to the second round of the game (which is 
).
Figure 6.3 Tree representation for the possible results of the game of craps with the probabilities for each of the come-out roll.
To complete the tree, we need the probability that the game stops conditionally on each of the six different points that can appear in the come-out roll. Consider first the probability of winning if the point is 4. We can break this event down into winning in the first point roll after 4 becomes the point, or the second point roll, or the third point roll, and so on. All of these events are disjoint, therefore
Now, the probability of winning in the first roll if the point is a 4 is simply the probability of getting a 4 in a roll of the dice
On the other hand, to win in the second roll if you did not win in the first and the point is four, your first roll must have been anything but a 4 or a 7 and your second roll should be a 4. The probability that the first roll is not a 4 or a 7 is 
, while the probability that the second roll is 4 is 
. Since the rolls are independent, this means that
A similar argument can be used for subsequent rolls. In general, the probability of winning in 
rolls if you have not won in the previous 
and the point is 4 requires that you observe a series of outcomes that looks like
where X corresponds to any outcome of the dice that is not a 7 or a 4. This sequence has probability
which leads to
Sums as the one in brackets are called geometric sums and appear very often when analyzing games that consist of sequences of independent trials. The following result is useful when dealing with geometric sums:
Finite Geometric Series
The sum of a geometric series with 
terms is given by
Note that, if 
and 
is very large then 
, which leads to
Infinite Geometric Series
The sum of an infinite geometric series is given by
Accordingly,
and therefore
The following R code can be used to verify the formula for the infinite geometric sum:
Another way to find the probability of winning when the point is 4 is to realize that, among the nine outcomes that end the game (three that add to 4 and six that add to 7), only the ones that add up to 4 make you win. This leads to 
, just as before. The following simulation can be used to corroborate the calculations we just made:
A similar argument can be used for all the other points (and a small modification of the code above can be used to check them, see Exercise 14):
Figure 6.4 Tree representation for the possible results of the game of craps with the probabilities for all scenarios.
The fully filled tree is presented in Figure 6.4. Now, following the Total Probability Law we discussed in Chapter 5, we can sum the probabilities associated with the paths that lead you to win the come-out roll to get
Hence,
which simplifies to 
. Furthermore, since the pass line bet pays even odds, the expected profit for every dollar invested is
As you can see, the house advantage for the pass line bet in craps is much smaller than the house advantage in roulette! On purely monetary terms then, there is no reason for you to ever play roulette again!
We can extend the code we used before to check the probability of winning given that the point is 4 to corroborate the probability of winning when playing the pass line bet.
6.1.2 The Don't Pass Line Bet
The don't pass line (also called lose or wrong) bet is a wager against the shooter that is almost the mirror image of the pass line bet and also pays even odds. The don't pass line bet is always played as a stand-by bet that runs parallel to the pass line bet and is resolved in the following way. If the come-out roll made by the shooter is a 7 or 11, then the don't pass line bet loses automatically. On the other hand, if the come-out roll is a 2 or 3, then the don't pass line bet wins automatically, but if the come-out is a 12 then the game ends in a tie (this is sometimes called a push) and the player gets her original bet back. Finally, if a point is made, the player betting on the don't pass line wins if a 7 comes up first and loses if the point comes up.
Note that, except for the outcome for 12 in the come-out roll, the don't pass line bet is the opposite to the pass line bet. Hence, we can easily compute the probability of winning and the probability of tying this bet using a similar procedure to the one outlined in the previous section. This leads to
and therefore
Note that the don't pass line bet is slightly less disadvantageous to the player than the pass line bet! You can modify the simulation of the pass line bet we provided in the previous section to check these results (see Exercise 16).
6.1.3 The Come and Don't Come Bets
In addition to the pass and don't pass line bets, there are two more line bets usually offered by casinos: the come and the don't come bets. A come bet works almost identically to the pass line bet, but it is played out of synchrony with and independently of it. As soon as the player makes a come bet, it starts its own first phase regardless whether the shooter is playing his come-out roll or a point roll. Consequently, if a 7 or 11 is rolled on the first round after the player placed the chips in the come area of the table, it wins, but if a 2, 3, or 12 is rolled, it loses. On the other hand, if the roll is 4, 5, 6, 8, 9, 10 then the come bet will be moved by the base dealer onto a Box representing the number the shooter threw. The don't come bet is similar, but it mirrors the don't pass line bet instead.
6.1.4 Side Bets
In addition to line and come bets, many casinos allow for single-roll and multi-roll bets. These bets can be placed at any time by either the shooter or any stand-by player. Examples of a single-roll bet include snake eyes (which involves betting on two ones coming up, typically paying 30 to 1) and the Yo (which involves betting on 11 coming up and often pays 15 to 1). The analysis of these bets is very similar to the analysis of the bets in roulette. For example, the expected profit from a snake eyes bets is
while the expected profit for Yo is
Note that the house advantage for these two bets is much larger than for the line bets available in craps (as well as for any of the bets available in roulette). Hence, these side bets are usually very bad proposition for the player.
An example of multi-roll bets is the hard way bet, in which the player bets that the shooter will throw a 4, 6, 8, or 10 the hard way (recall Table 6.1) before he throws a 7 or the corresponding easy way. We can use some of the ideas discussed in this chapter to compute house advantage for these bets. For example, in the case of the hard 8 bet (which pays 9 to 1 on a winning bet), note that there is only one dice combination that makes you win (two 4s), while there are 10 combinations that make you lose (6 ways in which 7 can happen, plus 3 and 5, 5 and 3, 2 and 6, and 6 and 2). Since any other number just forces you to continue rolling, the probability of winning this bet is 
, the probability of losing is 
, and the expected profit is
Again, this bet is quite bad for the players!
6.2 Exercises
-
1. When playing craps, what is the probability of crapping out?
-
2. When playing craps, if your point is 9, what is the probability that you will win within the next four shots? What is the probability that you will lose within the next four shots?
-
3. When playing craps, if your point is 5, what is the probability that you will win within the next four shots? What is the probability that you will lose within the next four shots?
-
4. If your point is 6, what is the probability that you will win the round of craps? Show your calculations.
-
5. If your point is 10, what is the probability that you will win the round of craps? Show your calculations.
-
6. If your point is 11, what is the probability that you will win the round of craps? Show your calculations.
-
7. Show that the probability of winning the don't pass line bet is
. -
8. Verify the value provided in the text for the house advantage in the don't pass line bet.
-
9. The don't pass line bet in craps is meant to be a bet against the shooter. Indeed, the don't pass line bet loses if the shooter gets a 7 or 11 in the come-out roll, or if the shooter gets the point during the follow-up rolls. However, the don't pass line bet wins in the come-out roll only if the shooter gets 2 or 3 but ties if the shooter gets a 12. Why are the rules setup in this way instead of just letting the don't pass line bets win for all come-out rolls that are 2, 3, or 12?
-
10. Why does a casino allow players to retire (take back) a don't pass line bet after the first roll has been made, yet it does not let you do the same for pass line bets?
-
11. What is the probability of winning the come and don't come bets?
-
12. In craps, the field bet is a single-roll wager in which the player wins if the next roll is 2, 3, 4, 9, 10, 11, or 12, and losses on any other number. The typical payout for this bet is 1 to 1 if 3, 4, 9, 10 and 11, 2 to 1 on a 2, and 3 to 1 on a 12. What is the house advantage for this bet, and how does it compare with the house advantage in the pass line bet?
-
13. What is the house edge on a hard 6 bet? How does it compare against the house edge for a hard 8 bet?
-
14. [R] Find out the value of the cumulative sums associated with the series
using R. -
15. [R] Create a simulation to compute the probability of winning the pass-line bet if the point is 9.
-
16. [R] Create a simulation to compute the probability of winning the don't pass line bet.