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How bet size and decision rate affect your gambling prospects

26 December 2011

Luck affects the outcome of any gamble. It may favor the well-prepared, but is still what gambling is about. At one extreme, you can count cards at blackjack and have an advantage over the house or flip coins for even money with a 50-50 shot and no edge, and lose your shirt on either. At the other end of the spectrum, you can buy a lottery ticket with faint hope of triumph and usurious edge, and become a multimillionaire. The math can’t predict whether you’ll win or lose, and how much. What it can do is determine your prospects, based on four parameters.

Average bet size.

Edge or house advantage. This gives the difference between the odds of success and the payoff. It’s normally expressed as a percentage and indicates the fraction of the total wagered over many trials that the casino gets to keep.

Volatility. This is most often measured by what the math mavens call standard deviation, which can be envisioned as the representative bankroll change on one or a series of coups.

Decision rate. Depending on how you choose to do the arithmetic, this is either the number of rounds per unit time dealt at a table or triggered by a player at a machine, or the number of bets that get resolved with a win or loss per unit time.

Consider a session on a typical slot machine. Say you bet $1 per spin on a machine with 94.5 percent player return; house advantage is the complementary 5.5 percent, such that the joint’s average profit over many trials is $0.055 per dollar bet. Assume the standard deviation is $7.16 per round. Both figures include allowances covering 1-for-1 returns, which are actually pushes, not wins. Punching the spin button on the average of once every 10 seconds yields six rounds per minute – 360 per hour. A spin every five seconds amounts to 720 rounds per hour.

Since edge moves money only one way, from bettors to bosses, it accumulates additively. The theoretical impact of a 5.5 percent edge at $1 per bet is therefore a loss of $1 x 360 x 0.055 or $19.80 after an hour with 360 spins; it’s twice as great, $39.60, after the same time span with 720 rounds. Volatility may raise as well as lower a player’s fortunes. The characteristic bankroll change during a series of bets increases as more wagers are made, but the growth is slower than that of edge. If the standard deviation of a single $1 bet is $7.16, that of 360 such wagers is $135.85 and of 720 rounds is up – not doubled, but by a factor of 1.4 – to $192.12.

Combined, average bet size, edge, and volatility establish a range in which solid citizens are apt to find themselves after any given number of decisions. In particular, the laws of probability indicate that 34 percent of all players should be within one standard deviation above and 34 percent one standard deviation below the average corresponding to the cost of the edge. Of individuals who get a spin every 10 seconds on the machine cited, this means that 68 percent would be between a profit of $116 and a loss of $155 after an hour. Analogously, of those who press the button once every five seconds, 68 percent would be from $152 up to $232 down.

Craps offers dice devotees some propositions with much lower edge and less volatility than those on most slot machines. With a few exceptions, though, the minimum bets casinos will book at this game are higher and the decision rates lower than those on the machines. Owing to the interactions among these factors, comparisons of the prospects for craps and for the slots are less straightforward then those who seek low edge on this and other table games tend to believe.

Consider a Place bet on an eight, for instance. The nominal edge is 1.52 percent, based on the five of 36 possible dice totals that win and the six that lose on this number. With the remaining 25 outcomes – effectively pushes – taken into account, the numerical value of edge on this wager falls to 0.46 percent; the standard deviation per dollar wagered, also predicated on 36 outcomes, is $0.596 per throw. The minimum Place bet on an eight is usually $6, and $12 is common. At $12, the casino’s theoretical take is 0.0046 x $12 = $0.055 and the standard deviation is $7.16 per throw. Both are the same as for $1 bets on the machine previously described.

The situations differ in decision rate, however. While 360 to 720 spins per hour are typical on the slots, craps tables rarely see over 100 throws per hour. At this rate, the effect of the edge on a bettor with a $12 eight is a theoretical loss of only $5.50 per hour. And the range represented by one standard deviation in this time span is $71.60 up or down. So, after an hour, solely with this wager, 68 percent of all players should be between about $66 to the good and $77 in the hole.

In general, you can therefore anticipate earning or losing more money on the slots than the tables. Not because the machines have inherently higher edge and volatility for ordinarily-sized bets, but owing to the differences in decision rates. As the punter’s poet, Sumner A Ingmark, penned:
While gambling speed may meet your need,
Playing faster courts disaster.

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.