🎲 Dice Games

Probability analysis in dice games (Monopoly, Craps)

Dice are one of the oldest random-number generators, known to humanity for thousands of years. Most board games use two standard six-sided dice (denoted as 2d6). The sum of the values ​​rolled on two dice can range from 2 to 12. However, a key characteristic is that this sum is not distributed unevenlu. Unlike a single die (1d6), where each number from 1 to 6 has an equal 1/6 probability of appearing, the sum of two dice results in different values ​​having different probabilities.

Why the uneven distribution?

The reason lies in the number of ways (combinations) to obtain a particular sum. There are 6 × 6 = 36 possible outcomes when rolling two dice (since each die is independent). Each combination (for example, (1,1) or (3,5)) is equally likely. However, different sums correspond to different numbers of combinations.

  • Total of 2: (1,1) → 1 way. Probability \( \frac{1}{36} \approx 2,78\% \).
  • Total of 3: (1,2), (2,1) → 2 ways. \( \frac{2}{36} \approx 5,56\% \).
  • Total of 7: 6 ways → \( \frac{6}{36} = 16,67\% \) (max).
  • Total of 12: 1 way → \( 2,78\% \).

Adapting into Monopoly

In Monopoly, players roll two dice and move forward by the number of spaces shown. Due to an uneven distribution, players land on some spaces significantly more often than others. This influences the strategy for purchasing property:

Spaces located 7 squares away from frequently visited areas (such as "Jail" or "Go") receive more traffic.

Spaces corresponding to the sums of 6 and 8 are also popular.

Spaces that can only be reached by rolling a 2, 3, 11 or 12 are visited infrequently and purchasing them is less advantageous.

Experienced Monopoly players know that the "orange" and "red" properties (in the classic version) are situated at an optimal distance from the most probable dice-roll totals, which is why they offer a better return on investment.

🎲 3D-Dice: Two Dice Roll
Total: 0 | Rolls: 0
Frequency of 7: 0%

A common misconception

One might think that, since a sum of 7 occurs most frequently, if a 7 has not appeared for a long time, it is bound to show up soon, which is a fallacy. Each roll is independent from any other, and the probabilities do not change. However, when making strategic decisions, it is important to consider the a-priori distribution rather than speculating about some sort of "compensation".

Expected value of the sum of two dice

The expected value (average result) of a roll of two fair six-sided dice is easy to calculate. For a single die, the expected value is: \[ \mathbb{E}[d6] = \frac{1+2+3+4+5+6}{6} = 3.5 \] Since the dice are independent, the expected value of their sum is equal to the sum of their expected values: \[ \mathbb{E}[d6 + d6] = 3.5 + 3.5 = 7 \] That is precisely why the sum of 7 is the most likely outcome — it coincides with the mean value. This property is used in many games for balancing purposes.

More formally speaking, for a discrete distribution: \[ \mathbb{E}[X] = \sum_{s=2}^{12} s \cdot P(X=s) = 2\cdot\frac{1}{36} + 3\cdot\frac{2}{36} + \dots + 12\cdot\frac{1}{36} = 7 \] Knowing the expected value helps players estimate their average winnings or progress across the board over a large number of turns.

Comparing with a single die: when is 2d6 better?

In some games, one needs to meet a certain threshold (for example, an ability check in Dungeons & Dragons). Rolling a 1d20 results in an even distribution, while rolling 2d6 produces a bell-shaped distribution centered at 7. The probability of rolling a 10 or higher on 2d6 is: \[ P(\ge 10) = P(10)+P(11)+P(12) = \frac{3}{36}+\frac{2}{36}+\frac{1}{36} = \frac{6}{36} \approx 16.7\% \] With a 1d20, the probability of rolling a 10 or higher is 55% (11 out of 20 possible outcomes). 2d6 yields much more predictable results, which is important in tactical games where players generally dislike extreme outcomes.

Expected values in Catan

In Catan, each hex tile marked with a number (from 2 to 12) yields a resource whenever that sum is rolled on the dice. The expected number of resources produced by a single tile per roll is equal to the probability of its number appearing. A tile marked with a 6 yields, on average, \( \frac{5}{36} \approx 0.139 \) resources, while a tile marked with a 2 yields only \( \frac{1}{36} \approx 0.028 \). Recognizing this, players compete to control the "good" numbers (6 and 8); they also take into account that two tiles marked with 6 or 8 together yield \( \frac{10}{36} \approx 0.278 \) resources per roll — nearly 10 times as much as a single tile marked with a 2.

Formula for average income per round: \[ \mathbb{E}[\text{resources}] = \sum_{\text{field } i} P(\text{number}_i) \times (\text{amount of resources from field}) \] The optimal strategy is to maximize this expectation, keeping in mind the diversity of resources.

Variance and risk: is a high EV always better?

In the game of Craps, a "pass" bet has an EV of approximately -1.4% (the house edge). However, some other bets offer a higher EV but also come with significantly greater variance. For instance, a "hardway" bet (wagering on a specific outcome) offers a large payout, but the probability of winning is very low. For a player with a limited bankroll, high variance can lead to a rapid loss of funds, even if the EV is not significantly negative.

A measure of dispersion is the variance of the sum of two dice. For a single die: \(\text{Var}(d6) = \frac{35}{12} \approx 2.917\). For the sum of two independent dice: \(\text{Var}(d6+d6) = 2 \times \frac{35}{12} = \frac{35}{6} \approx 5.833\). Root-mean-square deviation: \(\sigma \approx 2.415\). This means that in approximately 68% of cases, the sum will fall within the range \([7-2.415, 7+2.415] \approx [4.6, 9.4]\), that is, between 5 and 9, inclusive.

Large numbers law and dice

If one performs a large number of rolls (for example, 10,000), the arithmetic mean of the sums obtained will tend toward 7. This is illustrated by the interactive demonstration above: the more rolls one makes, the closer the frequency of rolling a seven gets to 1/6 and the closer the average value gets to 7. In the long run, casinos and game developers rely precisely on these mathematical expectations.

The practical conclusion is that in board games, where chance determines the course of play, an understanding of expected value and variance allows for more well-considered decisions, such as whether to take a risk for a large but infrequent payoff or to opt for a stable but less lucrative strategy.