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Best of Alan Krigman

Gaming Guru

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A guide to anticipating bankroll swings in games you haven't tried yet

10 September 2012

Exceptional instances or runs of good or bad luck can make or break a casino visit for anybody. Few players ever encounter either. Most tend to bet the same amount or press or regress wagers over a modest range from round to round. Their fortunes therefore typically rise and fall gradually, albeit at rates depending on the chances and amounts they win or lose in any coup.

Experience with particular games and propositions, bet sizes, and bankrolls may give solid citizens reliable intuition about how far and fast to expect profits or losses to mount under normal circumstances. But these instincts don’t necessarily transfer among gambles. Trial by fire to ascertain the ebbs and flows of unfamiliar situations can be costly, and small samples don’t dependably portend what would hold over the long term.

A better approach involves the bankroll and individual bets a person might risk and the edge and volatility of the action at issue. Single- and double-zero roulette can illustrate how these factors mesh. For uniformity, alternate bets totaling $12 with bankrolls of $120, $240, $360, $480, and $600 will be compared. The $12 will be wagered as $12 on one spot (pays $420), $6 on each of two spots (pays $204), $4 on each of three spots (pays $132), $3 on each of four spots (pays $96), $2 on each of six spots (pays $60), and $1 on each of 12 spots (pays $24). Edge is the same for all choices within the games: 5.26 percent for double-zero and 2.70 percent for single-zero. Volatility tracks the payoffs – highest for $12 on one spot and lowest for $1 on each of 12. And, to generalize, think of bankroll and bet size as a ratio going from 10/1 at $120 to 50/1 at $600.

The nearby tables provide probabilities of doubling a bankroll before busting out (I and II) and of not depleting a bankroll within 180 spins – three to four hours of play – (III and IV) for the two versions of the game. The data show that for either success criterion, cutting edge improves chances, but not by much. Higher bankroll-to-bet ratio has a large favorable impact on ability to survive downswings. It’s effect on likelihood of earning larger multiples of bankroll is adverse, with the influence stronger at lower volatilities. In and of itself, higher volatility raises the likelihood of reaching a win goal but lowers that of surviving a session of desired duration. And, in any case, raising chances of higher earnings lowers those of longer sessions, and vice versa.

Table I: Probability of doubling a bankroll before busing out at single-zero roulette

Bankroll   Spots among with $12 is evenly divided (volatility high –> low)					
             1      2      3      4      6      12
120        32.8%  32.3%  31.7%  31.1%  29.7%  24.7%
240        32.3%  31.2%  30.0%  28.8%  26.3%  17.5%
360        31.8%  30.1%  28.4%  26.7%  23.1%  11.8%
480        31.2%  29.1%  26.9%  24.6%  20.1%  7.7%
600        30.7%  28.1%  25.4%  22.7%  17.4%  4.9%

Table II: Probability of doubling a bankroll before busing out at double-zero roulette

Bankroll   Spots among with $12 is evenly divided (volatility high –> low)					
             1      2      3      4      6      12
120        32.3%  31.2%  30.0%  28.9%  26.3%  17.7%
240        31.2%  29.1%  26.9%  24.7%  20.2%  7.9%
360        30.2%  27.1%  23.9%  20.8%  15.0%  3.2%
480        29.2%  25.1%  21.1%  17.4%  10.9%  1.2%
600        28.2%  23.2%  18.6%  14.4%   7.7%  0.4%

Table III: Probability of not depleting a bankroll within 180 spins at single-zero roulette

Bankroll   Spots among with $12 is evenly divided (volatility high –> low)					
             1      2      3      4      6      12
120         9.5%  13.2%  16.0%  18.4%  22.4%  32.4%
240        18.9%  26.3%  31.9%  36.5%  44.4%  62.7%
360        28.2%  38.9%  46.8%  53.2%  63.4%  83.6%
480        37.1%  50.6%  60.0%  67.3%  78.0%  94.3%
600        45.6%  61.0%  71.2%  78.4%  88.0%  98.5%

Table IV: Probability of not depleting a bankroll within 180 spins at double-zero roulette

Bankroll   Spots among with $12 is evenly divided (volatility high –> low)					
             1      2      3      4      6      12
120         8.9%  12.1%  14.3%  16.2%  19.1%  25.5%
240        18.0%  24.4%  29.2%  33.1%  39.5%  54.3%
360        26.9%  36.6%  43.6%  49.3%  58.5%  77.5%
480        35.7%  48.1%  56.9%  63.7%  74.0%  91.3%
600        44.1%  58.7%  68.4%  75.5%  85.2%  97.3%

Edge or house advantage is usually well-publicized for table and video poker games. This is generally not true for non-poker slots; however, it’s reasonable to assume edge will be between 6 and 10 percent (94 and 90 percent player return, respectively). Volatility is another matter. Accepted measures of volatility can be calculated when the chance of and payoff for a win are known, but figures for this parameter are not widely circulated. For present purposes, it’s enough to assume that the lower the probability of a win and bigger the payoff, the higher the volatility.

Gamblers have some control over punting performance. At blackjack and video poker, players’ choices during a round influence edge. At craps, picking propositions on which to bet bears on edge and volatility. At roulette, spreading or concentrating bets affects volatility. And, in any situation, selecting bankroll-to-bet ratio impacts likelihood of meeting profit or survival goals. The uncertainty is in the randomness of individual events. Those who believe they exert control at this level missed what the bard of Avon, William Shakespeare, had Hamlet tell Horatio:

There's a divinity that shapes our ends/Rough-hew them how we will
and to which the bard of bettors, Sumner A Ingmark, added:
So the casinos’ profit often tends/to come from plans unfilled.

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.