Understanding the Mathematics of Personal Finance

PROBABILITY AND ODDS

Two equivalent measuring systems that are commonly used in gambling are the probability of winning (or losing) and the odds of winning (or losing).

If there are (W) ways of winning (say, the number of ways of pulling a green ball out of a box of 100 green and 300 red balls) and (L) ways of losing (the number of ways of not pulling a green ball, i. e., of pulling a red ball), then the probability of winning is the number of ways of winning divided by the total number of possible results. In the case of the box of 100 green and 300 red balls,

The odds of winning is the number of ways of winning divided by the number of ways of losing, usually expressed as a fraction with the word “to” designating division:

Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by

Lawrence N. Dworsky

Copyright © 2009 John Wiley & Sons, Inc.

Odds = 1:3 =

In the red/green ball example, the following four statements are equivalent:

1. The probability of winning is 0.25 = 25%.

2. The probability of losing is 0.75 = 75%.

3. The odds of winning are 1: 3, usually read as “1 to 3.”

4. The odds of losing (or against winning) are 3: 1, usually read as “3 to 1.”

Converting between the two systems is easy:

Odds = P: (1 - P) or

p = Odds = 1 + Odds.

To use the formulas, the odds must first be written as a fraction, for example, 1 :3 = 1/3. When calculating odds from a probability, you’ll get a fraction that will involve numbers less than 1. You have to scale the results up to integers before they look familiar. For example, aprobability of 1/3 gives odds of 1/3: (1 -1/3) = 1/3: 2/3. In this case, multiply the numerator and denominator by 3 to get the familiar result 1:2.

In common jargon, when the probability of success is 0.5 = 1/2, the odds are 1: 1, called “even odds.” When the probability of success is greater than 1/2, the first number in the odds is larger than the second number, for example, 3 : 2. This is called “the odds are with you.” In the opposite case, for example, 2 : 3, the probability is less than 1/2, and “the odds are against you.” When the probability of success is very small, the second number in the odds is much larger than the first number, for example, 2 :15, and the odds against you are “very long.”

In dealing with odds, 1: 3 is the same as 2: 6 and as 3: 9, and so on. While expressing probabilities as odds may seem more intuitive to some people, it’ s a somewhat limited system. A probability of 0.12 (a little smaller than 0.125 = 1/8) is clear and accurate, but the equivalent odds of 12: 88 = 3: 22 is an awkward repre­sentation. Something like 13:279 is really bad.