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MATH 1 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
Let p =
a n
be the probability that event A^ occurs, and let q =
b n
be the probability
that A does not occur, where a + b = n
(i) If p ≥
, then we define the odds in favor of A to be the ratio
a : b (which should be
simplified algebraically).
(ii) If p <
, then we define the odds against A to be the ratio
b : a (which should be
simplified algebraically).
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 ) =
(ii) P ( A ) = 0.32 (iii) P ( A ) =
(i) Out of every 7 attempts there should be 5 favorable occurrences and 2 non- favorable; 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 non- favorable; 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.
If
a : b are the odds in favor , then
a is the number of favorable outcomes and
b is the number of non-favorable. Then P ( A ) =
a a + b
If
c : d are the odds against , then the number
c coming first is the number of non- favorable outcomes. The second number
d is the number of favorable outcomes. Thus P ( A ) =
d 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 non- favorable, so P ( A ) =
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 P ( A ) =^
- What are the odds in favor of rolling an even number? ______________
- What are the odds against rolling an even number? ______________ number?
A sweepstakes has 500 entries. You have purchased one ticket. Calculate the following:
- What is the probability that you will win the 14. What is the probability that you will not win sweepstakes? the sweepstakes?
______________ ______________
- What are the odds in favor of you winning the 16. What are the odds against you winning the
sweepstakes? sweepstakes?
______________ ______________
Challenge questions:
- If you purchase two tickets in the sweepstakes, does that double the probability that you will win?
- If you purchase two tickets instead of one, use one of the following words to describe the likelihood of you winning the sweepstakes:
Certain, Impossible, Unlikely
Odds against or in favor
Practice:
In her wallet, Anne Kelly has 14 bills. Seven are $1 bills, two are $5 bills, four are $ bills and one is a $20 bill. She passes a volunteer seeking donations for the Salvation Army and decides to select one bill at random from her wallet and give it to the Salvation Army. Determine: a) The probability she selects a $5 bill b) The probability she does not select a $5 bill c) The odds in favor of her selecting a $5 bill d) The odds against her selecting a $5 bill
A box contains 9 red and 2 blue marbles and 3 yellow marbles. If you select one at random from the box, determine: a) The probability the marble is red. b) The odds in favor of selecting a red marble. c) The probability the marble is blue. d) The odds against selecting a blue marble
A pair of dice is rolled and the sum of the dice is recorded. Here is the sample space.
a) Find the probability of rolling a sum of 7. b) Find the probability of not rolling a sum of 7 c) Find the odds in favor of the sum being 7. d) Find the odds against the first dice showing a 5. e) Find the probability the sum is less than 7. f) Find the odds against of the sum being less than 7. g) Find the probability of rolling a double (both dice have the same number). h) Find the odds against rolling a double.
