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Best of Alan Krigman

Gaming Guru

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Expected Values Help You Juggle Payoffs and Probabilities

25 February 2003

Payoffs and probabilities in gambling are intimately intertwined. Ignore the coarse, if not the fine, structure of the connection at your peril. One extreme may have you pursuing prizes you'll reach only with phenomenal luck. The other, well, as Robert Burns warned, "the best laid plans of mice and men often go awry."

Picture the significance of the payoff-probability relationship in terms of a simple bet such as $10 on the nine at craps. A win pays $14, raising a purse from $10 to $24. The chance of winning is 40 percent. Is the payoff adequate? Is the chance reasonable?

Neither question stands on its own. But you can get an answer by multiplying the target bankroll times the probability, to get an "expected value" or "expectation." The product in this case, 40 percent of $24, is $9.60. That's the expected value of $10 when you bet on the nine at craps. So risking $10 to win $14 on the nine "costs" $0.40. And $0.40 is 4 percent of $10, a figure some solid citizens will recognize as the house advantage on the bet.

Expectation can also show the relative importance of alternate results in games having payoff schedules rather than one winning amount for each wager. The accompanying list gives returns, probabilities, and contributions to overall expectation for final hands in expertly-played "9/6" jacks-or-better video poker.


Expectations for various hands at 9/6 jacks-or-better
video poker (returns and expectations are per dollar bet)

Hand
Return
Probability
Expectation
Royal
800
0.00249%
$0.019920
Straight flush
50
0.01053
0.005265
Four of a kind
25
0.23640
0.059100
Full House
9
1.15200
0.103680
Flush
6
1.08000
0.064800
Straight
4
1.13500
0.045400
Three of a kind
3
7.44900
0.223470
Two Pair
2
12.94000
0.258800
High Pair
1
21.45000
0.214500
Lose
0
54.54458
0.000000
Total
100.00000
0.994935

 

Overall expectation is $0.994935. Edge equals about 0.5 percent, the theoretical loss ($1 - $0.994935 or just over half a cent) per dollar. Ranking, two pair adds the most to expected value, followed by triplets, high pair, full house, flush, quads, straight, royal, and straight flush. Were a full house worth only 8-for-1, its contribution to expectation would drop from $0.10368 to $0.09216 -- over a cent on the dollar -- boosting house edge more than 1 percent. If two pair returned 1-for-1, as in many "bonus" games, its share of expectation would fall from $0.2588 to $0.1294; this is nearly $0.13 on the dollar, a huge penalty compared to what's usually given back on the bonuses.

Expectation can also be determined for any given playing duration -- a session, casino visit, or even a lifetime. Bet a total of $1,000 at single-zero roulette, all at once or over some time span. The expected value of the $1,000 is $973. In essence, the casino "charged" you $27, 2.7 percent of $1,000, to play.

Pretend you're playing blackjack for $10 per round, on one spot, at a table with six other people. You'll get about 100 hands in two hours, risking a total of $1,000. If you follow perfect Basic Strategy, at 0.5 percent house edge, the overall expected value of the money you bet is $995. The casino takes a theoretical $5.

There's more. Expected values of wins and losses can be found separately. And the values afford insight into how well or badly you can anticipate doing by gambling in a certain manner.

For the same $1,000 handle, expectation on the plus side alone is $42.57 while on the minus it's $47.57. These figures suggest how far up or down you might be prepared to finish in a typical game this long with these bets. And, further, expectation peaks at $110.53 profit and $115.53 loss, suggesting how high or low you should be prepared to soar or plummet during a normal session.

However much or little you know, you can still win or lose. But understanding chances and amounts can help you establish goals and heighten the joy or temper the surprise when you hit or miss. The beloved bettors' bard, Sumner A Ingmark, said it succinctly:

Gamblers betting realistically,
Candidly but optimistically,
Set parameters statistically.

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.