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Best of Alan Krigman

Gaming Guru

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Playing It Smart: Do 10s follow 10s in blackjack?

19 November 2007

Some blackjack buffs override Basic Strategy with a superstition to the effect that "10s follow 10s." The gist of this is, 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 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).

Chance of various gaps between 10s for an infinite shoe

					simulations
	cards between	theoretical	10 MM	(A) 100	(B) 100	(C) 100
	successive 10s		hands	hands	hands	hands
	0	30.77%	30.52%	34.00%	23.00%	38.00%
	1	21.30%	21.70%	23.00%	23.00%	21.00%
	2	14.75%	15.00%	12.00%	22.00%	25.00%
	3	10.21%	10.04%	12.00%	8.00%	7.00%

Even with 10 million hands, where simulation results should converge on the theoretical averages, some offset from the expected values is apparent. With 100 hands, departures from the predictions caused purely by randomness and not by any fundamental occult or other driving forces are more evident.

A person tracking simulation (C) would observe 38 percent of all draws showing 10 followed immediately by another 10, and a total of 84 percent where 10s had no more than two intervening cards. Relying on this sample as proof, a gambler might be strongly tempted to argue that "something was going on" to make 10s follow 10s, and perhaps to bet on this secret the casino bosses didn't want anyone to know. An individual counting only 10s followed immediately by 10s, and seeing the hands of simulation (B), would wonder how this wacky idea ever originated.

So, does something about blackjack make 10s follow 10s? Yes, the distribution of ranks in a standard deck gives this a probability of 30.77 percent. Will this be the frequency you obtain in any particular group of tries? No, you could be under or over as a result purely of the mixing of the cards during the shuffle. The data from the simulation make this clear.

Finally, does knowing the probability provide information beyond what's inherent in Basic Strategy that can help you improve your game? Don't bet on it -- figuratively or literally. For, as that wily wordsmith, Sumner A Ingmark, was want to warn:

If you see one swallow and bet it's summer,
Don't try blackjack using this rule-of-thumb or,
Your casino visit will be a bummer.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.