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MAKE 10s FOLLOW 10s?

By Alan Krigman

Alan Krigman, and his poetic sidekick Sumner A. Ingmark, have been illuminating the dark recesses of casino gambling for more than a dozen years. Mr. Krigman is especially well known for sharing his insights into the mathematics underlying the various games (including blackjack), the influence of volatility and skewness as well as edge on bankroll during the course of a session, and the impact of betting as well as decision strategies on expected performance. A searchable archive of Mr. Krigman's prose and Mr. Ingmark's muse is online at


Some blackjack buffs override Basic Strategy with a superstition to the effect that "10s follow 10s." The gist of this is as follows: if a 10-value appears, the next card is more apt than by chance to be another 10.

One explanation given by players, who want to think they have a sound basis for beliefs that seem to fly in the face of reason, involves the way cards are racked after each hand and subsequently shuffled. Another is that it's simply among those mysteries of gambling destined to forever elude mere mortals like wins and losses occurring in streaks, and people who hit big jackpots always having premonitions that it's gonna happen that day.

The laws of probability suggest why solid citizens, who assume 10s follow 10s, will be reinforced in their conviction by what they see in practice. Ignoring shoe size and card withdrawal, which won't affect the numbers too much, there are four 10-values out of every 13 cards in a standard array. So prospects for a 10 on any card are four out of 13, or 30.77 percent. Bettors can, therefore, expect to see 10s follow 10s in almost a third of all cases. They'll be off by one card a 10 followed by a non-10 then by a 10 with a probability of 21.30 percent. Together, on the average, these represent 52.07 percent over half of all draws. To be sure, the same could be said for any non-10 followed by a 10. But nobody's superstitious about something like 10s having a special propensity to come after fives, so who'd notice?

Everybody knows, of course, that just because the chance of an event is four out of 13, the phenomenon of interest isn't ordained to occur exactly four times in every series of 13 trials. You might not get the target result at all, or you might find it 13 times running. This is what a random process, which is what shuffling is supposed to achieve, is all about.

The effect is shown in the accompanying table. The data give the percentages of instances in which a 10 is followed immediately, and separated by one, two, or three cards, by another 10. Columns indicate theoretical values derived from the laws of probability, as well as results of computer simulations of 10 million hands and three arbitrary sets of 100 hands each (labeled A, B, and C)...

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