What Happens to Chance when a Game Isn't Random?

Casino games are often described using "cycles." Few folks still take the idea literally, picturing predetermined sequences comprising all possible results. Most understand cycles as aids in applying the laws of probability to gambling.

For instance, a slot machine with four 10-stop reels has 10,000 positions, so a cycle is 10,000 numbers. The probability of any given value on a spin is one out of 10,000. However, the machine isn't programmed to produce each outcome once every 10,000 spins.

Likewise, craps is often characterized using 36-throw cycles. In these, each two-die combination occurs the number of ways it can be formed (once for a two, twice for a three, ... six times for a seven, and so forth). Similarly, for many purposes, 13 cards is considered a cycle. Here, depending on the game, each rank surfaces once, or 10-values pop four times and the rest once.

Ordinarily, a real sequence whose length equals that of one or a few "cycles" is highly unlikely to exhibit the exact theoretical distribution. For example, the chance of a "perfect" 36-roll dice series is a mere one out of 706,154. Even if such a sequence does transpire, having the "correct" instances of each result is only part of the story. The arrangement of the events in the string also matters. In fact, order may have more impact on gambling performance than frequencies of occurrence. The phenomenon is enigmatic because "order" is defined uniquely for each game.

This isn't an issue at the slots. Each spin is a new trial. The random number generators (RNGs) in state-of-the-art machines not only yield all possible numbers with uniform probability, they do so in a manner that meets statistical criteria for randomness.

In other games, order - even in a perfect cycle - may spell the difference between success and failure. For instance, gamblers familiar with craps could deduce that the following perfect cycle of 36 throws, 4-4-10-4-10-5-10-5-5-5-9-2-3-3-12-9-9-9-6-6-6-6-8-8-8-8-7-7-7-7-11-11-7-7-6-8, would be worth $180 to someone making only a single flat $10 bet on the pass line, starting the set with a shooter's come-out roll. The same solid citizen would lose $100 after encountering this probabilistically-correct sequence: 4-7-2-3-3-12-4-11-11-10-7-4-5-6-8-9-10-7-5-6-7-5-6-6-9-9-8-9-8-7-5-10-6-8-8-7. How many perfect 36-roll sequences would be good or bad for some style of play? It'd take a long time to work out the answer; over 432,000,000,000,000,000,000,000,000,000 different arrangements are possible.

Likewise, pick all the clubs from a single deck of cards. With faces defined as equivalent 10-values, as at blackjack and baccarat, you can arrange the set 259,459,200 different ways. The distribution of ranks is perfect because you're using a complete suit. But, were the same 13 cards dealt in a game, wins or losses can diverge significantly from one permutation to the next.

In a four-position round of blackjack, 2-4-A-3-5-7-6-8-10-10-10-9-10 would earn bettors a total of $80, assuming $10 wagers and Basic Strategy; 2-4-A-10-6-8-7-9-10-10-10-3-5 would cost the same players a total of $80. Both 13-card sets are perfect, having four 10-values and one of everything else. The order makes the difference. And, the devil is in the details. A winning sequence with four spots in action might be a loser with three or five. And, what's great at blackjack might be awful at Let-It-Ride.

Authoritative gambling strategies tend to account for the probabilities of various events, and assume that whatever happens does so randomly. "Secret systems" (the kind casino bosses are said to dread, but can be mailed to your home in plain brown wrappers for $29.95) often involve streaks, clumps, and patterns. That is, they depend on how outcomes are ordered. But what, if anything, imposes order on nominally random processes? And, must you understand the mechanism to take advantage of it? Questions, not answers. But, just knowing the questions is a step forward. This is what the poet, Sumner A Ingmark, meant when he mused:

In earnest quest for wisdom, the student's most arduous task,
May not be finding answers, but learning there's something to ask