How To Measure Chance: Odds, Fractions, and Percentages

28 October 2003

By Alan Krigman

It's happened. You're a heartbeat away from a royal at video poker. The machine, the gods, or a classicist might say both in the form of a deus ex machina, dealt you the jack, queen, king, and ace of clubs ?? along with a four of spades ?? for starters. Not to toss too wet a blanket on your excitement, but how likely are you to cop the biggie or one of the consolation hands?

To find out, count the "ways" you can reach levels of joy. Then figure the chances by accounting for how many cards are left in the electronic "deck" from which you'll draw to replace the four. But, how do you combine the numbers to get probabilities?

Say it's a jacks-or-better game, nothing wild, a 52-card deck. Here are the hands that get you something back, and the number of cards that'll make 'em. The lowest is a high pair: 12 ways (any of the three remaining jacks, queens, kings, or aces). Next, a non-royal straight: three ways (the 10 of hearts, diamonds, or spades). Then comes a non-royal flush: eight ways (deuce through nine of clubs). On top is the royal: one way (the 10 of clubs).

Maybe the game has one joker, a 53-card deck. The lowest winner is a non-royal straight: three ways (the 10 of hearts, diamonds, or spades). Next is a non-royal flush: eight ways (deuce through nine of clubs). Finally, usually, two different royals, with the joker or the 10 of clubs: one way each.

To figure your chances, associate ways to win with either the opportunities to fail or the universe all possible results. And express the relationship as odds, fractions, or percentages.

When a proposition is adverse, odds are stated as "(number of ways to fail)-to-(number of ways to succeed)." Under favorable circumstances, the terms flip and odds are given as "(number of ways to succeed)-to-(number of ways to fail)." You can go further and "normalize" the odds to help reconcile what otherwise might be apples and oranges. Divide the larger number by the smaller to get "quotient-to-1." Take the jacks-or-better non-royal flush as an example. After the initial round you've seen five cards, leaving 47 in the deck. The eight ways to hit leave 47 - 8 or 39 to miss so the odds are 39-to-8, normalized as 4.875-to-1.

The fraction is "(the number of ways to succeed) out of (the total number of possibilities)." This can also be normalized by dividing the larger number by the smaller to get "one out of quotient." In written form, you often see these expressed as (number of ways to succeed)/(total number of possibilities) or 1/quotient. For the jacks-or-better non-royal flush, the fraction is "eight out of 47" (8/47) or "one out of 5.875" (1/5.875).

The percentage is equivalent to the fraction, but requires the arithmetic to divide (number of ways to succeed) by (total number of possibilities). The jacks-or-better non-royal flush yields eight divided by 47 or 0.1702, which is 17.02 percent.

The accompanying chart shows the options for a one-card draw to a single-ended possible royal in both games, to one decimal place.

	normalized		normalized
	odds	odds	fraction	fraction	percent
J-B pair	35-to-12	2.9-to-1	12/47	1/3.9	25.5%
J-B str	44-to-3	14.7-to-1	3/47	1/15.7	6.4%
J-B flush	39-to-8	4.9-to-1	8/47	1/5.9	17.0%
J-B roy	46-to-1	46.0-to-1	1/47	1/47.0	2.1%
J-W str	45-to-3	15.0-to-1	3/48	1/16.0	6.3%
J-W flush	40-to-8	5.0-to-1	8/48	1/6.0	16.7%
J-W J-roy	47-to-1	47.0-to-1	1/48	1/48.0	2.1%
J-W roy	47-to-1	47.0-to-1	1/48	1/48.0	2.1%

The alternatives are equivalent. When using odds and fractions, most solid citizens find the normalized forms more intuitive, especially when contrasting different situations. For instance, in thinking about non-royal straights and flushes in the joker-wild games, 15-to-1 and 5-to-1 are easier to compare than 45-to-3 and 40-to-8. Percentages are toughest to comprehend but provide the most uniform comparisons. With fractions, they also offer the advantage of allowing direct calculations of quantities even more useful than merely the chance of winning. But, that opens a whole new can of worms. Since, as the poet, Sumner A Ingmark, wrote:

Percentages, precise indeed,
Though basic to what gamblers need,
Are often twisted to mislead,
And can't predict when you'll succeed,
Or warn when sense gives way to greed.