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How Long Must You Play before the Law of Averages Takes Effect?2 October 2000
Everyone knows that casinos have the edge on every bet they book at their machines or tables. A few grizzled gamblers also realize that in the statistically short span of a typical session, the inherent volatility of casino games eclipses this house advantage. So, virtually anything can -- and does -- happen.
Some players win big despite the edge. Others lose much more than the experts explain they should expect. Such results are the rule, not the exception. And they not only occur in games where performance is influenced by playing and wagering strategies, but also where there are no options but bet or walk away.
What about the legendary "long run," though? If you keep playing, will the casino bosses eventually get you for exactly, or nearly, the product of the edge times the total of your bets?
To understand how, and how much, chance continues to intervene, eliminate house advantage and consider a gamble based only on volatility. To do so, make believe you're in a game comprising "fair" bets -- wagers in which neither side has an edge. Examples might range from flipping an unbiased coin for even money to dropping $1 on a million-to-one shot with a $1,000,000 payoff.
Because there's no edge, in principle, you should neither win nor lose in the long run. "Expectation" is zero. However, you could certainly encounter intervals during which you were ahead or behind. So, imagine you hit a cold streak when you start. If you're willing and able to plough onward, can you rely on the law of averages to get you back?
I'll explore this issue from two perspectives, illustrating with the flip of a coin. Fair bets with odds other than 1-to-1 work similarly, but need more trials to reach comparable points.
The first element of interest is whether your chances of being near the expected value -- break-even in a fair game -- really improve as the duration of a session lengthens. Most solid citizens believe that the theory dictates a narrowing of the gap, bringing you closer to home, as play proceeds. Does it?
Players usually picture wins or losses in terms of a specific sum, say $10, up or down. After 100 flips of the coin at $1 per decision, your chance of being $10 or less above or below break‑even is 68.3 percent. Go 1,000 rounds and the probability drops to 24.8 percent. After 10,000 and 100,000 tries, the likelihood you'll be no more than $10 ahead or behind is 8.0 and 2.5 percent, respectively. Your chances of finishing within any fixed band around the expected value get worse rather than better, the longer you play
The second key item of importance is the degree to which playing longer affects the likelihood that if you get into a hole early in a game, you'll recover your losses. This suggests a different question with respect to having the bankroll, time, and temperament to extend the action. Namely, do your prospects of never being even or ahead decrease with number of rounds? Here, the intent is to grind it out until you finally return to ground zero or show a profit after being behind, then quit.
The answer is encouraging, but needs to be interpreted correctly. Flip a coin once; it's obvious that you have a 50 percent shot at being behind. Play for 10 rounds and the chance you'll never be even or ahead drops to about 25 percent. After 100 trials, the probability falls to about 8 percent. In 1,000 attempts, it's only 2.5 percent. And, if you can endure 10,000 rounds, you've got under 1 percent chance. The longer you're prepared to play, the less you're apt never to be even or ahead during -- but not necessarily at the end of -- the game.
Notice that in both instances, we're discussing chances growing or shrinking. Nobody mentioned guarantees. At least, nobody who ever heeded this advice from the poet, Sumner A Ingmark:
A sign of gamblers immature,
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