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Best of Alan Krigman

Gaming Guru

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What Do You Do when Chance Says "No" but Expectation Says "Yes"

25 September 2001

Normal folks talk about bets in terms of the chance they'll win, lose, or maybe push. Chance is expressed numerically as the odds or probability of an event. Gurus often discuss "expected value" or "expectation." These give theoretical profit or loss assuming a statistically-correct distribution of results over time.

Chance and expectation are related. But they're distinct concepts and the difference between them can be, if not profound, at least perplexing. In particular, one wager or tactic may be more apt to win than another, yet have less expectation. Or conversely.

The disparities grow from two roots. 1) Payoffs that involve high multiples, outright or varying with specific outcomes achieved, so returns counterbalance chances. 2) Bets that can push as well as win or lose, so net rather than gross determines expectation.


As an example of the first factor, say you have 7-8-9 of hearts in Let It Ride, and the first community card is the 10 of hearts. The hand will be resolved by the second community card, one out of the 48 left in the deck. This could yield a straight flush with a six or jack of hearts (two cards), a flush with any heart other than the six or jack (seven cards), a straight with a six or jack of any suit other than hearts (six cards), a high pair with a 10 (three cards), or a loss with anything else (30 cards).

A four-card outside possible straight flush is obviously strong. Still, it crashes a lot more than it flies. Of the 48 unknowns, 18 win at least even money and 30 lose. That's 18/48 = 37.5 percent chance to win and 30/48 = 62.5 percent to lose.

The strength of the hand isn't in chance of winning but in expectation. The following chart should help clarify this point.

Payouts, probabilities, and expectations
for a suited 7-8-9-10 at Let It Ride
resolution payout probability expectation
Str Flush 200 4.17% 8.3333
Flush 8 14.58% 1.1667
Straight 6 12.50% 0.7500
High pair 1 6.25% 0.0625
Loss -1 62.50% -0.6250
Expectation.............................................................. 9.6875

The first three columns show resolutions, amounts won or lost per dollar bet, and probabilities. The last column is the probability multiplied by the amount. Expectation is found by totaling the fourth column ?? essentially adding the chances of winning times the amounts won, then subtracting the probability of losing times the amount lost. For the values shown, expectation works out to a healthy theoretical profit of just under $9.69 per dollar bet.

A 12 against a dealer's 2-up in blackjack shows how pushes can cause expectation to go one way and chance the other. A final 12 wins only if the dealer breaks. Drawing a 10 busts. Other draws win if the dealer breaks or ends below the player's total.

The chance that a dealer will bust with 2-up is 35.3 percent. Therefore solid citizens who stand on 12 versus 2-up should win 35.3 percent and lose the other 64.7 percent. A 12 can't push, but only win or lose. The amount will be the same, one unit either way. So expectation is 35.3 - 64.7 = -29.4 percent. That's $0.294 per dollar bet, the minus sign auguring a loss.

Players are projected to win 34.8 percent of the time by hitting. This is four bets per thousand worse than the 35.3 percent by standing. However, 5.0 percent of all cases should push. The chance of a loss is accordingly 100 - 34.8 - 5.0 = 60.2 percent. Expectation for this situation is 34.8 - 60.2 = -25.4 percent, a theoretical loss of $0.254 per dollar bet. This is a $0.04 per dollar improvement over by standing.

What's best when there's an option? Expectation offers greater long-run promise. But chance has merit here and now. Few players would consider taking back their second bets at Let It Ride on a suited 7-8-9-10 because the chance of winning is less than that of losing. But an abundance of blackjack buffs balk at hitting 12 versus 2-up. The beloved bard, Sumner A Ingmark, saw both sides:


Two ducks on a pond are what hunters set mind on,
But one in a pot is more readily dined on.

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.