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Best of Alan Krigman

Gaming Guru

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Chance to Win Depends on Fluctuation as well as Edge

12 March 1996

Casino bets have only two inherent variables: the probabilities and amounts players can win or lose on each decision. It should therefore be easy to examine every game, analyze these two factors for all bets and decision strategies, and deduce what and how to play to get the greatest chance of making money.

It should be easy. But it isn't. Although the two variables are each simple, they're related in a complex way. That relationship, along with the number of bets made during a session, governs the ultimate likelihood and extent of profit or loss.

Most gaming experts -- consultants who confer with casinos as well as specialists who suggest strategies to solid citizens -- combine probabilities and payoffs into a single parameter, "edge," characterizing each bet. Edge is like a fee, a percentage the casino taxes every wager. It can be below 0.5 percent of routine blackjack and baccarat bets. In double-zero roulette, edge is more than 5 percent. Propositions -- the "center" bets -- in craps go over 16 percent. In sic-bo, edge can exceed 18 percent. And some state lotteries grab about 50 percent. The higher the edge, the more advantage the casino has on the action. But, edge is only half the story.

A second parameter, "variance" -- or, in handier form, "fluctuation" -- can also be found from the probabilities and payoffs. Fluctuation (purists will recognize it as normalized "standard deviation") is measured in betting units. It describes the volatility of a game; the higher the fluctuation, the greater the up- and downswings players are likely to experience during a series of bets. Fluctuation is close to 1.0 on bets that pay even money nearly half the time, rises to 5 or 10 on longshots in table games, and can go through the roof on "jackpot-only" slot machines.

Volatility is rarely considered in assessing bettors' chances. But fluctuation as well as edge are necessary to ascertain the probability that players will win or lose any specified amounts. And, both parameters are essential to determine how much bankroll is required to survive normal downswings and how big a win can be reasonably expected after some number of bets. As an example of the need for the two factors, the probability of just being ahead falls with edge and rises with fluctuation, remaining the same if the pair change in exact proportion to one another.

In extreme cases, edge can be ignored and only fluctuation used to establish the probability of a result. This occurs, for instance, on slot machines which emphasize large but infrequent payoffs. Even with edge as high as 10 or 15 percent, where the casino's fee is 10 or 15 cents for every dollar bet, the chances that a player with a reasonable stake will boom or bust can be calculated solely from fluctuation and number of tries.

The accompanying table gives some specific examples of how players' expectations vary with edge and fluctuation in roulette and craps. The table shows the probability that a player will lose or win more than the indicated amount after 100 decisions with each of four different bets.

You can draw several important conclusions even from the limited data presented in the table.

  1. You're always more likely to lose a specified sum than to win the same amount; the effect worsens as edge increases.
  2. Regardless of edge or fluctuation, the greater your win goal or loss limit, the lower the probability you'll reach it.
  3. For any edge, your chance of making a big score or of suffering a heavy loss both rise as fluctuation gets higher.
  4. Edge drops in relative importance as fluctuation increases.

Edge works only one way -- for the casino. Fluctuation can help as well as hurt a player, although the house is still favored because it has the bankroll and time to continue no matter how far it gets behind or how long it takes to recover. As artfully articulated by Sumner A Ingmark, the bettors' bard:

On the upswing, I said "I won't quit yet."
On the downswing, I had no more to bet.

TABLE
Probability of a loss or win greater than the indicated amount
after 100 decisions for four representative bets

roulette:roulette:craps:craps:
$5 on a single $1 each on five$5 on the ten;$5 on the nine;
number; numbers;
edge=5.26%edge=5.26%edge=6.67%edge=4.00%
fluctuation=5.81fluctuation=2.43fluctuation=1.32fluctuation=1.17
loss over $20027.6%7.7%0.7%0.1%
loss over $10039.827.215.98.7
win over $046.441.430.736.7
win over $10033.114.92.12.0
win over $20021.93.10.00.0
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.