In the Casino, Seeing Shouldn't Always Be Believing

Theories abound as to what makes cards, dice, and pictures on slot machine reels and screens exhibit the trends and patterns many bettors know they see and believe occur. Gamblers reinforce their beliefs by watching for supporting instances, taking these events as evidence and ignoring everything else.

Among my favorite examples is the notion that "10s follow 10s" in blackjack. That is, if a 10-valued card appears, the next card is more likely than usual to be another 10. Carried further, this theory implies that runs or clumps of successive 10s occur more often than predicted strictly by the laws of probability.

Bettors who believe 10s follow 10s are sure they're right because they see it with their own eyes. They've heard that chances of five, six, or more 10s in a row are minuscule. But they've played plenty of blackjack and noticed numerous runs like these. And dealers tell them they observe it "all the time." Some solid citizens embrace the hypothesis without trying to explain it, just as they accept microwave cooking without wondering what gets the potato hot. Others think 10s follow 10s because of imperfect shuffles as well as the order in which cards are picked up and placed in the discard rack during and after a round.

It's a fact that the chance of drawing a long series of successive 10s is quite small. It's also a fact that players -- and to a greater extent, dealers -- will see them fairly often. These two facts aren't contradictory. Taken together, they indicate why occurrences of clumps won't help beat the odds.

To picture what I mean, assume cards are drawn from an infinitely large shoe. The laws of probability state that the chances a random draw will yield four 10s in a row are roughly 9 out of 1,000. The chances of five and six 10s in a row are under 3 out of 1,000 and 1 out of 1,000, respectively. Pretty low.

So, assume that in the course of an hour's play, you observe a run of six or more 10s in a row. And, further, that you notice runs at least this long almost every time you play. Wouldn't you be tempted to conclude that since the chances of such a thing are so small, your observing it regularly proves something about 10s following 10s and the clumping of cards in the shoe?

You'd be wrong. During an hour, you're apt to see 1,250 cards. And a run of six 10s could occur anywhere in that set of cards -- from positions 1-2-3-4-5-6, 2-3-4-5-6-7, 3-4-5-6-7-8, through 1243-1244-1245-1246-1247-1248, 1244-1245-1246-1247-1248-1249, 1245-1246-1247-1248-1249-1250. The chance of seeing that 1 out of 1,000 event is increased by the possibility that it can occur at any of 1,245 positions. Added to this, you'd likely believe the theory verified if you observed sequences such as five rather than six 10s in a row, or six out of seven successive cards being 10s. All told, the chance of seeing something you think confirms the theory can get fairly high, simply by virtue of the many ways for the individually rare events to occur and independently of anything special about 10s following 10s or clumping.

Here are some specific figures for perspective. The chance of at least one clump of 10s longer than five cards in a series of 1,250 completely random draws is almost 40 percent. That is, you can expect to see one or more such runs in four out of every 10 hours you play. The chance of seeing at least one run longer than six cards in an hour of play exceeds 14 percent. And the chance of one or more runs over seven cards in length is over 4 percent.

How can this information help you? It arm you with knowledge. By rejecting spurious theories, you're better able to resist the urge to do something silly after seeing a few 10s in a row, falsely anticipating an unusually high chance of drawing 10 again. Long runs of 10s are random phenomena, which are likely during a game by chance alone. No clumping mechanism induces them to happen. You therefore can't predict when they'll occur or how long they'll last. As the poet, Sumner A Ingmark, whose random runs of rhymes never forecast what's to follow, reminded readers:

Those too quick to be believing,
Are no one but themselves deceiving.