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Best of Alan Krigman

Gaming Guru

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Playing It Smart: Why the most likely result of your gambling isn't what you expect

12 August 2008

Gambling gurus often rank wagers by their "expected value." It's a complementary measure of house advantage. When a casino has 2 percent edge, expected value is (100 - 2) or 98 percent of a bet. Think of it like this. At 2 percent edge, the bosses earn an average of 2 percent of the gross wager $0.02 on the dollar. At 98 percent expected value, the players finish with an average of 98 percent of their action $0.98 on the dollar. The same $0.02, gain or loss, depending on your perspective.

Imagine an even-money proposition with 49 percent chance of ecstasy and 51 percent of agony. Make believe you bet a dollar on it 100 times. In a statistically-correct universe, 49 coups would win $1 and 51 would lose $1. You'd be down by $51 - $49 or $2 on a $100 handle. This is 2 percent the bosses hold the edge. You began with $100 and finished with $98 so your 100 $1 bets were worth an average of $0.98 each the expected value.

There's a fly in the ointment, however. Solid citizens betting $10 on a round in some Byzantine manner could conceivably end with $9.80. They're likelier to conclude with $20, $30, ... $100, or more or with zero. Ditto for a session. A slot buff may play a 98 percent return machine once-through with $100 and cash out with $32, $257, $728, or whatever or leave empty-handed; $98 is possible, but isn't favored in any particular way.

Why the disparity between expected value and the actual result of a gamble? It's because expected value in probability theory is an artifice. It describes the quality of a wager in general, but doesn't relate to any particular situation. Who'd be so foolish as to lay down a buck confident of getting only $0.98 back?

In the 49-51 even-money example, a minority of 49 rounds would return $2 while a majority of 51 would render nothing. The "most likely value" of the dollar bet by a 2-round margin is therefore zero. And, most likely value, a metric little noted in even the most erudite gambling circles, is arguably far more significant to individual casino aficionados than expected value.

The contrast between expected and most likely values heightens as payoffs rise and chances fall. Picture a jackpot-only slot machine with 9,999-to-1 odds of losing $1 or winning $9,999. A model 10,000 statistically-correct rounds would break even 9,999 $1 losses and one $9,999 win. So a $1 bet has $1 expected value. But the most likely value, by 9,999-to-1, is zero.

Perhaps you consider a buck as chump change when it buys a shot at $9,999. So "utility theory" says you accept the overwhelming prospect you'll leave with zip, and don't care whether the chance of winning is one in 10,000 or 25,000 the expected value of a $1 bet in these cases being $1 or $0.40, respectively.

Expected and most likely values tend to approach one another when numbers of trials are statistically large. Make 1,000 $1 bets in the 49-51 percent even-money game. The expected value of the $1,000 gross wager is $980 490 rounds in which your $1 becomes $2. This will also equal or be close to the most likely value.

But "statistically large" is relative. Suppose you start with $1,000 and take 1,000 $1 pulls at the hypothetical jackpot-only full-pay machine. The expected value of your stake at the conclusion of the session is $1,000. But you have 90.5 percent chance of going the 1,000 rounds without a hit. So, by odds of 90.5-to-9.5, the most likely value of your $1,000 is zero.

There's something else. Nothing close to either what's expected or most likely in a casino will change your life for good or bad. So, other than for entertainment, why go at all? And why learn to play proficiently? The answer lies in "the Black Swan" effect a phenomenon outside of normal expectation, with major implications, that seems logical after the fact but can't be predicted before. More on Black Swans another time. For now, just ponder this profound pronouncement of the poet, Sumner A Ingmark:

An average is useful statistically,
To show events characteristically,
But may not mean much realistically.

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.