Probability Stated as Odds

Probability Stated as Odds.

Given a probability P(E),

Odds for E = P(E)/[1 − P(E)].

Given odds for E of “a to b,” the implied P(E) is a/(a + b).

Odds against E = [1 − P(E)]/P(E)

Given odds against E of “a to b,” the implied P(E) is b/(a + b).

If P(E) = 1/8, the odds for E are (1/8)/(7/8) = (1/8)(8/7) = 1/7, or “1 to 7.”

In wagering, for odds of “15 to 1” against E 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

[PRACTICE PROBLEMS] If the probability that Zolaf Company sales exceed last year’s sales is 0.167, the odds for exceeding sales are closest to:

A.  1 to 5.

B.  1 to 6.

C.  5 to 1.

[Solutions] A

Given odds for E of a to b, the implied probability of E = a/(a + b).

Stated in terms of odds a to b with a = 1, b = 5, the probability of E = 1/(1 + 5) = 1/6 = 0.167. This result confirms that a probability of 0.167 for beating sales is odds of 1 to 5.