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Best of Alan Krigman

Gaming Guru

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Law of Averages Repealed; Casino Players Win and Lose

7 May 1996

You flip coins for a buck a shot with your chums. You figure it's a harmless pastime because the law of averages says you'll come out even over a long period. Being a good scout, though, you keep records for tax purposes and find, after 5,000 flips, you're $50 - maybe $100 - ahead.

You also relish the casino experience and are a frequent patron. You exercise good judgement as to types and amounts of bets, loss limits, and size of wins at which you quit. You've educated yourself about gambling and know the casino's statistically-expected "hold" on your action equals your cumulative bet times the small house edge. And the law of averages says you'll only lose the expected value over a long period - for instance, $5 or so per hour with good basic strategy at $10 blackjack or about $4 per hour at craps when placing the six and eight for $6 each. You consider this a fair price for the pleasure you derive. Being a good scout, though, you keep records for tax purposes and find that after a year playing a few hours of $10 blackjack a week, you're $500 - maybe $1000 - ahead.


What's going on? Are the mathematicians wrong? Is the coin you flip or the casino game you play biased in your favor? Are you one of those supernaturally lucky solid citizens? Has the new Republican Congress repealed the law of averages?

The math mavens are right, although they're often misinterpreted. And results like those suggested are normal in unbiased situations, regardless of the good or bad luck you may think you may have. As for the law of averages, there never really was any such thing to repeal. The valid laws, of probability and statistics, yield an "expected value" but don't ordain your getting there; actually, they predict the likelihood you won't.

The accompanying table shows what can happen by flipping a coin for a dollar four times, giving results as W or L - win or lose. Four flips can be resolved 16 distinct ways. Each set of outcomes is equally likely and, together, yield a zero average or expected value. But consider the 16 possibilities: six break even for a probability of 6/16 or 37.5 percent, four win $2 for a probability of 4/16 or 25 percent, and one wins $4 for a probability of 1/16 or 6.25 percent. Adding these probabilities shows a 68.75 percent chance of being even or ahead by any amount and 31.25 percent chance of being $2 or more ahead. The same math, of course, yields a 68.75 chance of being even or behind by any amount and 31.25 percent chance of being $2 or more behind.


Putting worse lie to the law of averages, the more you flip, the lower the prospect of breaking even and the higher that of having a big win or loss. Compare 1,000 and 5,000 flips at $1 each. Chances of ending up precisely at zero are 2.52 and 1.13 percent, respectively. Chances of being even or ahead by any amount are 51.26 and 50.656 percent for the two cases. To be up $20 or more, however, the respective probabilities are 27.44 and 39.375 percent. For $50 or more, they are 10.94 and 24.325 percent. At the $100 win level, chances are a mere 0.17 percent after 1,000 flips and a much greater 7.965 percent after 5,000 tries. The same numbers, to be sure, also apply to $20, $50, and $100 losses.

Similar reasoning holds on casino games, with two key modifications. 1) In flipping coins, deviations are amounts over or under expected values of zero; in casino games, deviations are departures from expected losses due to house advantage. 2) The statistical fluctuation in flipping coins is a single bet unit; in the casino, it varies from just over a unit in even-money table games to 30 bet units or more on table and slot machine longshots. And, high fluctuations mean more chance of being further above or below expected values after any number of tries, as well as asymmetrical situations where players' bankroll limitations introduce factors like a few big wins balancing many small losses.

Sumner A Ingmark, bard beloved of bettors because his verse verbalizes vagaries they venerate, voiced the verity vibrantly:


A check of the data savages,
Belief in the law of av'ages


TABLE
Possible results
of flipping an
unbiased coin
four times
W W W W
+4
W W W L
+2
W W L W
+2
W L W W
+2
L W W W
+2
W W L L
0
W L L W
0
L L W W
0
W L W L
0
L W L W
0
L W W L
0
W L L L
-2
L W L L
-2
L L W L
-2
L L L W
-2
L L L L
-4
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.