Understanding Craps Probabilities

Craps is a fast‑paced casino classic built around the simple act of throwing a pair of fair six‑sided dice. Despite the straightforward equipment, the game supports a wide array of wagers whose odds differ markedly. A working knowledge of those odds is essential for evaluating the risk behind every cheer at the table. The calculator above focuses on five staples: the Pass Line, Don't Pass, Field, and Place bets on the numbers six and eight. When you pick one of these wagers, the script computes the probability of a win and the corresponding house edge, offering a concise statistical snapshot before you join the next roll.

At the heart of craps lies the distribution of sums from two dice. Because each die has six faces, there are $6\times 6=36$ equally likely outcomes. The sums range from two to twelve, but not with equal frequency. For example, a total of seven can be produced in six different ways, while a total of two occurs in only one way. The table below lists every possible sum with its count of combinations and exact probability.

Sum	Combinations	Probability
2	1	1/36
3	2	2/36
4	3	3/36
5	4	4/36
6	5	5/36
7	6	6/36
8	5	5/36
9	4	4/36
10	3	3/36
11	2	2/36
12	1	1/36

The Pass Line is the quintessential craps wager. It is resolved in two stages. On the opening throw, called the come‑out roll, a total of seven or eleven wins immediately, while two, three, or twelve loses. Any other total establishes a point, and the dice are rolled repeatedly until either that point reappears (a win) or a seven surfaces (a loss). Using conditional probability, the chance of a Pass Line win can be expressed as:

$P\left(\mathrm{win}\right)=\frac{8}{36}+{\displaystyle \sum _{p\in \{4,5,6,8,9,10\}}\frac{c_p}{36}\frac{c_p}{c_p}}$

Here $c_p$ denotes the number of combinations that produce point p and $c7$ equals six. Plugging in the values gives a win probability of $\frac{244}{495}$, which is approximately 0.49293. The house edge for a one‑unit wager is therefore $\frac{7}{495}$ or about 1.41%.

The Don't Pass bet is the mathematical mirror of the Pass Line. On the come‑out roll, two or three wins, seven or eleven loses, and twelve pushes. If a point is established, the bet wins when a seven appears before the point repeats. The probability of winning becomes $\frac{251}{495}$ with a house edge of roughly 1.36%. Many players consider Don't Pass socially awkward because it wagers against the shooter, but statistically it offers a slightly better expectation.

The Field bet is a one‑roll proposition covering totals of two, three, four, nine, ten, eleven, and twelve. It loses if five, six, seven, or eight appears. Most casinos pay even money on winning numbers except two and twelve, which typically return double. Some houses triple the twelve instead. The probability of any field number is $\frac{16}{36}$. Accounting for the bonus payouts yields a house edge near 2.78% when twelve pays triple and 5.56% when it pays double. Because the wager resolves in a single roll, the variance is high.

Place bets allow you to directly back a point number after the come‑out roll. Betting on six, for instance, wins whenever a six is thrown before the next seven. In terms of combinations, six can be made in five ways while seven appears in six ways. The probability that six arrives first is therefore $\frac{5}{11}$. Casinos pay 7:6 on a Place 6 wager, implying a house edge of roughly 1.52%. The same logic applies to Place 8, which also has five winning combinations and pays 7:6, producing an identical edge.

The table below compiles the probabilities and house advantages for the bets handled by the calculator. The Expected Value column shows the average gain or loss per unit wager, calculated as $\mathrm{EV}=P\left(\mathrm{win}\right)\times \mathrm{Payout}-(1-P(\mathrm{win}\left)\right)$. A negative EV reflects the house edge.

Bet	Payout	Win Probability	House Edge	Expected Value
Pass Line	1:1	244/495	≈1.41%	≈−0.0141
Don't Pass	1:1	251/495	≈1.36%	≈−0.0136
Field (12×3)	1:1 triple on 12	16/36	≈2.78%	≈−0.0278
Place 6	7:6	5/11	≈1.52%	≈−0.0152
Place 8	7:6	5/11	≈1.52%	≈−0.0152

Knowing these figures can sharpen your strategy. A player focused on minimizing loss might prefer the modest edge of Don't Pass, while someone seeking quick excitement may gravitate toward the volatile Field bet. Regardless of approach, remember that no wager overcomes the fundamental disadvantage embedded in the game. The house edge is the casino's long‑term profit margin, and it ensures that even skillful play results in an expected loss over time. Used responsibly, the calculator provides clarity without encouraging overconfidence.

The math behind the Pass Line bet illustrates how conditional probability intertwines with combinatorics. After a point is established, the game becomes a race between that point and seven. If the point is six, for instance, the probability that six appears before seven equals the ratio of their respective combination counts. More formally, $\frac{c_6}{c_6}$. Because $c_6=5$ and $c_7=6$, the value becomes $\frac{5}{11}$. We multiply this conditional probability by the chance of establishing the point in the first place, $\frac{5}{36}$, to contribute $\frac{25}{396}$ to the overall win probability. Summing over all possible points completes the calculation.

Historically, casinos adopt payoff ratios that create tidy house edges. If Place bets on six and eight paid 6:5 instead of 7:6, the player would enjoy a positive expectation. Such payout adjustments are rare because casino managers calibrate them carefully. The uniformity of payoffs across establishments facilitates mathematical analysis and allows players to memorize optimal strategies.

Craps also provides a vivid demonstration of the law of large numbers. Individual sessions may end in big wins or losses, but as the number of rolls grows, the proportion of outcomes converges toward the probabilities shown above. A Pass Line player who somehow wins ten come‑out rolls in a row has defied the odds but not changed them. The expected value remains negative no matter the recent streak. By internalizing that truth, players can maintain perspective and approach the game with measured expectations.

Whether you are a seasoned dice controller or a curious newcomer, this calculator aims to illuminate the numerical skeleton beneath the exuberant surface of craps. By quantifying win rates and house edges for key wagers, it invites informed risk assessment. Treat the results as a guide rather than a guarantee, set firm limits on losses, and remember that the most reliable way to leave a casino with a small fortune is to arrive with a large one.