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# How Do You Evaluate Chance in Gambling

19 January 2001

Gamblers, and most civilians as well, are continually confronted with predicaments posed by uncertainty and chance. Yet, these phenomena are as elusive as they are pervasive. At the root of the puzzlement is the discomfort people experience with risk rather than assurance. Will it rain tomorrow, or not? We're happier with yes or no than maybe, even when the maybe is tagged with a level such as 4 on a scale of 17 where 1 means definite sunshine and 17 "Noah, launch the ark!"

Adding to the aggravation, two different yardsticks are used to describe and measure chance: probability and odds. More, within each category, numbers are assigned in several alternate ways. So, when a figure -- rough or of great precision -- is given for the chance of a proposition, what does it tell you?

In the casino, the most straightforward circumstances involve the chances associated with individual bets. Here, values are found by counting the numbers of ways different results can occur. As an illustration, say you're betting a single spot at double-zero roulette. The ball can stop in any of 38 distinctly numbered and colored but otherwise identical pockets. Nothing favors one over another. You win with one and lose with the remaining 37.

Probability expresses chance by comparing the number of ways you can win to the total range of possibilities. In this example, it's one out of 38. Mathematically, this is the fraction 1/38. If you have a calculator or remember long division, you could reduce 1/38 to the decimal 0.0263 or the percentage 2.63 percent.

Odds quantify chance by comparing the number of ways you can win to the number of ways you can lose. The indicated roulette bet has one winner and 37 losers. This can be stated as odds of 37-to-1 "against" the wager. An alternate would be 1-to-37 "for" the bet or, dusting off your arithmetic again, 0.027-to-1 for the proposition -- obtained by dividing 1 and 37, in turn, by 37.

These all describe the same chance. And the samples above show how to convert from one format to another within either category. How about switching from probability to odds, and vice versa?

Assume the odds of a bet are 9-to-4 against winning. This happens to describe "insurance" at blackjack -- assuming cards drawn from an infinite deck -- since nine (A through nine) lose while four (10, J, Q, and K) win. To convert to probability, note there are four ways to succeed out of four plus nine or 13 possible outcomes. This is 9/13, 0.6923, or 69.23 percent.

Going the other way is just as easy. Pretend the probability of an event is 40 percent. This turns out to be the chance of winning a place bet on the five at craps. What are the odds? A 40 percent probability is 40 ways to win out of 100 possibilities. This means 100 minus 40 or 60 ways to fail, so the odds are 40-to-60 for or 60-to-40 against the proposition. If you prefer, divide both sides of the former by 60 to get 0.667-to-1 for, or both sides of the latter by 40 to get 1.5-to-1 against the event. The odds against winning this bet are sometimes stated as 3-to-2, which is the same as 1.5-to-1 with both sides multiplied by 2.

Chance in gambling isn't limited to situations where numbers of ways to succeed or fail can be counted in advance. For instance, the chance you'll survive a session of some specified duration can be calculated from your bankroll, bet size, and strategy in any game. Suppose the probability is 42.7 percent -- 42.7 out of 100 or, perhaps more intuitively for this case, 427 out of 1,000.

One interpretation would be that for every 1,000 solid citizens who give it a shot, 427 should prevail and 573 bust out, on the average. Alternately, you can think in terms of the odds you'll deplete your bankroll within the allocated period. The figure is 573-to-427, or perhaps 1.34-to-1 if you divide both sides by 427. One way or another, the poet, Sumner A Ingmark, embraced such uncertainty in his oft quoted quartet:

Why is it the sure things,
Are often such poor things,
While odds I must batter,
For earnings that matter?

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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.