Odds

Let the probability of the occurrence of an event be denoted by p, thereby implying that 1 – p represents the probability of the nonoccurrence of the event. Succinctly, the odds of the occurrence of the event are defined as the ratio of the probability of the occurrence to the probability of the nonoccurrence, or

.

Odds are often encountered in gambling situations. A pertinent example can be extracted from a football game. For instance, if a Vegas odds maker assigns an 80% probability of a football team from western New York winning a match against another team from eastern Massachusetts, the suggested odds of the football team from western New York winning are

.

If the odds can be represented in terms of a fraction

, where a and b are both integers, the odds are often referred to as “a-to-b.” Thus, the odds in favor of the team from western New York winning the match would be quoted as “four-to-one.”

In the context of gambling, a “fair game” is one in which both the house and the gambler have an expected gain of zero. Indeed, if the odds maker sets the odds of the team from western New York winning at four-to-one, and the perceived odds among gamblers are consistent with this assignment, then the average gain for both the house and the gamblers should be near zero. In this setting, if a gambler has the discretionary income to bet $1 in favor of the eastern Massachusetts football team, and the team from eastern Massachusetts wins the match, the gambler would receive 4 × $1 = $4 in earnings plus all monies with which the bet was made ($1), thus accumulating a total of $5. Conversely, should the gambler choose to bet $1 in favor of the western New York football team, and the team from western New York prevails, the gambler would receive ¼ × $1 = $0.25 in earnings plus all monies with which the bet was placed ($1), thus accumulating a total of $1.25. Neither the gambler nor the house experiences a positive expected gain because the odds of the team from western New York defeating the team from eastern Massachusetts directly offset the difference in payouts.

In biostatistics, odds are often used to characterize the likelihood of a disease or condition. For example, if 100 people within a small village of 1,100 people contract a certain disease, then the odds of a randomly selected resident contracting the disease are

. An interpretation of this measure would be that the probability of becoming infected with the disease is 0.1 times the probability of not developing the disease.

Care must be taken to distinguish between odds and probability, especially when the measures are both between 0 and 1. Consider, for instance, tossing an unbiased, six-sided die. The probability of rolling a 1 is ⅙ however, the odds of rolling a 1 are ⅕, or “one-to-five.”

For any event that is not impossible, the odds are always greater than the probability. The previous examples illustrate this fact. The probability of a western New York football team victory is set at 0.8, whereas the odds are 4.0. The probability of a villager contracting the disease is 0.091, whereas the odds are 0.1. The probability of rolling a 1 in a die toss is 0.167, whereas the odds are 0.2.

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