A grossly oversized die, in particular, is quite vulnerable to such manipulation. The standardized size of dice used in casinos may well represent a compromise configuration—based on long experience—that maximizes the opportunity for fairness. Casinos and gambling regulations specify the ideal dimensions and weight of dice.

*The size of a die can affect how easily it can be manipulated. The two red dice (right) were precisely machined for casino use.*

A cheat can also doctor a die to increase the probability of or even guarantee certain outcomes. References to "loaded" dice, in which one side is weighted so that a particular face falls uppermost, have been found in the literature of ancient Greece, Nowadays casino dice are transparent to reduce the chances of such a bias being introduced.

*Casino operators detect loaded dice by dropping them carefully into a glass of water. A loaded die will tend to turn while descending through the water, whereas a fair die will sink with little rotation. These computer simulations show a fair die tumbling on all six sides (top), a loaded die constrained to tumble on just four sides (middle), and a fair die constrained to tumble on four sides. The motion of a loaded die tends to be slightly more erratic than that of a fair die.*

Even without deliberately creating a bias, it's difficult to manufacture dice accurately without introducing some asymmetry or nonuniformity. Manufacturers of casino dice take great pains to assure quality.

Typically 0.75 inch wide, a die is precisely sawed from a rectangular rod of cellulose or some other transparent plastic. Pits are drilled about 0.017 inch deep into the faces of the cube, and the recesses are then filled in with paint of the same weight as the plastic that has been drilled out. The edges are generally sharp and square.

In contrast, ordinary store-bought dice, like those used in children's games, generally have recessed spots and distinctly rounded edges. Because much less care goes into the fabrication of such dice, they are probably somewhat biased.

Achieving fairness is even more difficult with polyhedral dice that have eight, twelve, or twenty faces, each of which must be manufactured and finished to perfection.

*A 12-sided (or dodecahedral) die.*

In principle, an unbiased cubic die produces six possible outcomes. It makes sense to use a mathematical model in which each face has an equal probability of showing up. You can then calculate other probabilities, including how often a certain number is likely to come up.

Several decades ago, statistician Frederick Mosteller had an opportunity to test the model against the behavior of real dice. A man named William H. Longcor, who had an obsession with tossing dice, came to him with an amazing offer to record the results of millions of tosses (see "A Passion for Tossing Dice").

Mosteller accepted the offer, and some time later he received a large crate of big manila envelopes, each of which contained the results of 20,000 tosses with a single die and a written summary showing how many runs of different kinds had occurred.

"The only way to check the work was by checking the runs and then comparing the results with theory," Mosteller recalled. "It turned out [Longcor] was very accurate." Indeed, the results highlighted some errors in the then-standard theory of the distribution of runs.

Because the data had been collected using both casino dice from Las Vegas and ordinary store-bought dice, it was possible to compare their performance not only with theory but also with each other and with a computer that simulated dice tossing.

As it turned out, the computer proved to have a poor random-number generator (see "Random Bits"), whereas the Las Vegas dice were very close to perfect in comparison with theory.

A mathematical model allows us to analyze the behavior of dice in both the short term and the long run and to study how the randomness of tumbled dice interacts with the rules of various games to favor certain strategies and results.

In some versions of Chutes and Ladders (or its Snakes and Ladders counterpart), each player must roll a six in order to start the game and then roll again to make his or her first move. It may actually take a while for that particular number to materialize, but more often than not each player will succeed in four or fewer throws.

The mathematics of chance reveals why. On each of four rolls of a single die, the probability that six will come up is 1/6. Because each roll of a die is independent of any other roll, the probability that six will not come up in four rolls is 5/6 ✕ 5/6 ✕ 5/6 ✕ 5/6, or 625/1296. Hence, the chance of getting a six is 1 − 625/1296 = 671/1296 = .5177, a probability of more than 50 percent.

During the 17th century, a favorite gambling game in the salons of Paris involved betting with even odds that a player would throw at least one six in four rolls of a single die. The calculated probabilities demonstrate that anyone who made the bet could win handsomely in the long run, especially if the unsuspecting victim believed intuitively that it would take six rolls to get the desired result.

One variation of the game involved rolling a pair of dice to obtain a double six in 24 throws. In this case, the gambler would lose in the long run if he offered even money on the possibility.

One of the players who were puzzled by such an outcome was writer Antoine Gombaud, Chevalier de Méré (1607-1684). An old gamblers' rule said that two dice come up in six times as many ways as one die cast. Thus, if four is the critical number of throws in a game with one die to reach favorable odds, then six times four, or 24, would be the critical number of throws in a game with two dice.

Gombaud suspected that 24 wasn't the right answer, and he worked out an alternative solution to the problem, looking at the 36 different throws possible with two dice. He wasn't entirely sure of his conclusion, however, so he asked his acquaintance Blaise Pascal (1623-1662), a mathematician and philosopher, to check the reasoning.

Pascal determined that the probability of rolling a double six after 24 throws turns out to be less than ½. No double six comes up in 35 out of the 36 possible outcomes for two dice. Throwing the dice 24 times means that the probability of not getting a double six is 35/36 multiplied by itself 24 times, or (35/36)

^{24}= .5086.So, the probability of winning is only 1 − .5086, or .4914. Clearly, betting with even odds on such an outcome becomes a losing proposition in the long run.

*Possible results of throws with two dice. Rolling a fair die has six possible outcomes. In rolling two dice, there are six possible outcomes for the second die, which gives a total of 36 pairs. What's the probability of rolling two dice to obtain, say, five? Because the total comes up in four different pairs (shaded),the probability is 4/36, or 1/9.*

Even with one die, however, it often takes a player many more than four throws to roll a particular number. On one occasion in five, a player will not succeed within 12 throws, and on one occasion in a hundred within 24 throws of the die.

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**Rolls and Flips**Next:

**Climbing and Sliding**
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