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How often do various outcomes occur in games of chance?

3 December 2012

Casino players often wonder about the prospects of experiencing various numbers of rounds before some outcome occurs or between results. The event of interest may be desired or dreaded, depending on the situation. As examples, craps aficionados might want to see lots of numbers before a dreaded seven, roulette enthusiasts might hope that hits on a single-spot, a four-spot corner, or some other combination occur in close proximity to one another, and blackjack buffs might wish for runs packed with natural 21s or opportunities for Basic Strategy splits or doubles.

The question is often phrased in terms of average – more technically, “most likely” or “expected” – numbers of coups. One set of answers can be obtained directly from the probability of the condition. Since the chance of a seven at craps is one out of six, this total will befall on the average of once every six rolls. Similarly, at double-zero roulette, because the likelihood of any spot is one out of 38, a number will score on the average of once every 38 spins; for a four-spot corner, the probability and average rate of hits are four out of 38. Averages are less obvious when the odds are expressed as percentages. At eight-deck blackjack, prospects of Basic Strategy doubles are 9.64 percent; a bit of long division (1 divided by 0.0964) shows that this percentage equals one out of 10.4 or, if you prefer whole numbers to those pesky decimals and don’t mind a little more arithmetic, 5 out of every 52 hands. Inquiring minds might also want to know that for sanctioned splits, the probability is 2.56 percent, which equals one out of 39.1 or 10 out of every 391 hands. And, for natural 21s, it’s 4.74 percent or one out of every 21 hands.

These figures aren’t as predictive as the terminology implies. Few players gamble enough for results of their bets to home in on long-term averages. Moreover, the statistically expected value is merely more probable than other numbers of outcomes – but not necessarily by much. For instance, with sevens at craps, expectation is for one occurrence in six rolls. But chances of fewer or more sevens in six tries are: zero – 33.490 percent, one – 40.188 percent, two – 20.094%, three – 5.358 percent, four – 0.804 percent, five – 0.064 percent, and six – 0.002 percent.

Solid citizens may find themselves better prepared for what they consider hot or cold runs in the course of a game by knowing the chances of experiencing series of coups between results of interest. Representative figures are presented in the nearby table. Here are some examples of interpreting the data. The probability of eight craps throws during which no seven appears is 23.3 percent. The probability of 21 double-zero roulette spins yielding no hits on specified four-spot corners is 9.7 percent. And the probability of 50 blackjack hands not calling for a split under Basic Strategy is 27.3 percent.

Probabilities of series of coups during which indicated outcomes do not occur

    craps         00 roulette     00 roulette     blackjack     blackjack     blackjack	
  no seven        no 1-spot        no 4-spot      no double     no split    no natural 21	
coups   prob    coups   prob    coups	prob   coups   prob   coups   prob   coups   prob
  2    69.4%     10    76.6%      3    71.6%      3   73.8%     10   77.2%      5   78.4%
  4    48.2%     20    58.7%      6    51.3%      6   54.4%     20   59.5%     10   61.5%
  6    33.5%     30    44.9%      9    36.8%      9   40.2%     30   45.9%     15   48.3%
  8    23.3%     40    34.4%     12    26.3%     12   29.6%     40   35.4%     20   37.9%
 10    16.2%     50    26.4%     15    18.9%     15   21.9%     50   27.3%     25   29.7%
 12    11.2%     60    20.2%     18    13.5%     18   16.1%     60   21.1%     30   23.3%
 14     7.8%     70    15.5%     21     9.7%     21   11.9%     70   16.3%     35   18.3%
 16     5.4%     80    11.8%     24     6.9%     24    8.8%     80   12.6%     40   14.3%
 18     3.8%     90    9.1%      27     5.0%     27    6.5%     90    9.7%     45   11.2%
 20     2.6%    100    6.9%      30     3.6%     30    4.8%    100    7.5%     50    8.8%

With some eyeball interpolation, the values in the table can also be utilized to estimate how many coups in a row are needed before the probability of occurrence drops below some level. Say you want to know the number for roughly 50 percent probability. For no sevens at craps, the figure would be between three and four throws. For spins at double-zero roulette, it would be about 26 for misses on a single spot and 6 on a four-number corner. And at blackjack, numbers of hands would be approximately seven for no doubles, 26 for no splits, and 14 for no natural 21s.

• The inverse information is also enlightening. An illustration might be the probability of two or three of any specified results in a row. The probabilities are:one out of 36 or one out of 216, respectively, for two or three sequential sevens at craps;
• One out of 1,444 for hitting two straight-up bets and 16 out of 1,444 (four out of 361) for two four-corner wagers in tandem at double-zero roulette;
• One out of about 108 for two and one out of 1,116 for three successive doubles, one out of 1,526 for two and one out of 59,605 for three splits, and one out of 445 for two and one out of 9,39 for three successive natural 21s at blackjack.

Ultimately, of course, nobody gives expected frequencies of events much thought if a game is hot and the dough comes cascading in. And players get no comfort by knowing how big a stake the laws of probability said they needed to ride through the normal waits, when the games are cold and they go belly-up. Still, as the renown rhymster, Sumner A Ingmark, cagily cajoled:

Don’t be leery of the theory,
Or what fries you may surprise you.
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.