Odds in Favor or Against

Odds in Favor or Against Is there a difference in meaning between the phrases “One in a Hundred” and “A Hundred to one Shot”? The first phrase should be interpreted as a probability 1/100. That is, out of every 100 attempts, there should be 1 occurrence. However, the second phrase is stated as a ratio 100 : 1, and should therefore be interpreted as odds. We formally define “odds in favor” and “odds against” as follows: Fraction Version a b be the probability that event A occurs, and let q = be the probability n n that A does not occur, where a + b = n Let p = 1 , then we define the odds in favor of A to be the ratio a : b (which should be 2 simplified algebraically). 1 (ii) If p < , then we define the odds against A to be the ratio b : a (which should be 2 € simplified algebraically). (i) If p ≥ € Example 1. What are the appropriate odds of drawing a Club from a deck of cards? Solution. The probability of drawing a Club is p = 1/4 which is less than 0.50, so we should state the odds against drawing a Club. Since p = 1/4, then q = 3/4. Therefore, the odds of drawing a Club are 3 : 1 against. (Because p = 1/4, then for every 4 attempts there should be 1 favorable occurrence and 3 non-favorable; hence, the odds are 3 : 1 against.) Example 2. What are the odds of drawing from Two through 10 from a standard deck of 52 cards (excluding Jokers)? Solution. There are 9 desirable values from 13 possible, so p = 9/13 is the probability of success and q = 4/13 is the probability of failure. Thus, the odds are 9 : 4 in favor. Decimal Version Let p = P(A) be the probability that event A occurs, and let q = 1− p be the probability that A does not occur. (i) If p ≥ 0.50, then we define the odds in favor of A to be the ratio p : q (which should be simplified algebraically). (ii) If p < 0.50, then we define the odds against A to be the ratio q : p (which should be simplified algebraically). Example 3. In the state, 72% of registered voters are Democrats. If a registered voter is chosen at random, what are the odds of choosing a Democrat? Solution. Because p = 0.72 which is more than 0.50, we should state the odds in favor of choosing a Democrat. So the odds are p : q = 0.72 : 0.28 = 72 : 28 = 18 : 7 in favor. Example 4. Find the appropriate odds if (i) P(A) = 5 7 (ii) P(A) = 0.32 (iii) P(A) = 2 . 13 (i) Out of every 7 attempts there should be 5 favorable occurrences and 2 nonfavorable; hence, the odds are 5 : 2 in favor. (ii) Here p = 0.32, thus q = 1 – 0.32 = 0.68. Because the chance q of not happening is larger, the odds are q : p = 0.68 : 0.32 = 68 : 32 = 17 : 8 against A happening. (iii) Out of every 13 attempts there should be 2 favorable occurrences and 11 nonfavorable; hence, the odds are 11 : 2 against. Converting Odds to Probabilities The odds are always stated as a simplified ratio a : b , where a and b are positive integers and a ≥ b . (The larger number comes first.) Think of the sum a + b as the total number of possibilities. € € a : b are the odds in favor, then a is the number of favorable € outcomes and b is the If a number of non-favorable. Then P(A) = . a+b € € If c : d are the odds against, then the number c coming first is the number of nonfavorable outcomes. The second number d is the number of favorable outcomes. Thus d . P(A) = c+d € € € € Example 5. The odds of event A are 13 : 3 in favor. What is P(A) ? Solution. There are (13 + 3) = 16 possibilities, of which 13 are favorable and 3 are non13 13 = . favorable, so P(A) = 13 + 3 16 Example 6. The odds of event A are 15 : 9 against. What is P(A) ? Solution. Out of every (15 + 9) = 24 attempts, 15 are non-favorable and 9 are favorable, so 9 9 = . P(A) = 15 + 9 24