In business and elsewhere, we often encounter probabilities stated in terms of odds—for instance, “the odds for E” or the “odds against E.” For example, as of November 2013, analysts’ fiscal year 2014 EPS forecasts for JetBlue Airways (NASDAQ: JBLU) ranged from $0.55 to $0.69. Suppose one analyst asserts that the odds for the company beating the highest estimate, $0.69, are 1 to 7. Suppose a second analyst argues that the odds against that happening are 15 to 1. What do those statements imply about the probability of the company’s EPS beating the highest estimate? We interpret probabilities stated in terms of odds as follows:
Probability Stated as Odds. Given a probability P(E),
Odds for E = P(E)/[1 − P(E)]. The odds for E are the probability of E divided by 1 minus the probability of E. Given odds for E of “a to b,” the implied probability of E is a/(a + b).
In the example, the statement that the odds for the company’s EPS for FY2014 beating $0.69 are 1 to 7 means that the speaker believes the probability of the event is 1/(1 + 7) = 1/8 = 0.125.
Odds against E = [1 − P(E)]/P(E), the reciprocal of odds for E. Given odds against E of “a to b,” the implied probability of E is b/(a + b).
The statement that the odds against the company’s EPS for FY2014 beating $0.69 are 15 to 1 is consistent with a belief that the probability of the event is 1/(1 + 15) = 1/16 = 0.0625.
To further explain odds for an event, if P(E) = 1/8, the odds for E are (1/8)/(7/8) = (1/8)(8/7) = 1/7, or “1 to 7.” For each occurrence of E, we expect seven cases of non-occurrence; out of eight cases in total, therefore, we expect E to happen once, and the probability of E is 1/8. In wagering, it is common to speak in terms of the odds against something, as in Statement 2. For odds of “15 to 1” against E (an implied probability of E of 1/16), a $1 wager on E, if successful, returns $15 in profits plus the $1 staked in the wager. We can calculate the bet’s anticipated profit as follows:
Win: | Probability = 1/16; Profit =$15 |
Loss: | Probability = 15/16; Profit =–$1 |
Anticipated profit = (1/16)($15) + (15/16)(–$1) = $0 |
Weighting each of the wager’s two outcomes by the respective probability of the outcome, if the odds (probabilities) are accurate, the anticipated profit of the bet is $0.
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