Grade by Grade
A. Counting and Probability · Probability

Odds in Favour and Odds Against

A1.1 A1.3About 60 minutes
Learning goals
  • I can find the odds in favour of and the odds against an event.
  • I can convert between probability and odds in both directions.
  • I can explain the difference between odds and probability in everyday and media statements.
  • I can compare payout odds with true odds to judge whether a game is fair.

Words to know

complement (A′)
The event "A does not happen". P(A′) = 1 − P(A). (Lesson 7 studies complements in detail.)
odds in favour of A
The ratio P(A) : P(A′). For equally likely outcomes, it is (outcomes in A) : (outcomes not in A).
odds against A
The ratio P(A′) : P(A), the reverse of the odds in favour.
payout odds
The ratio a game or bet uses to pay winners, such as "pays 5 to 1". It may differ from the true odds.
fair game
A game in which, on average, no player gains or loses money: the payout odds equal the true odds against winning.

Another way to describe chance

Probability compares the favourable outcomes with all outcomes. Odds compare the favourable outcomes with the unfavourable ones.

Roll a fair die. Rolling a 6 has 1 favourable outcome and 5 unfavourable outcomes.

  • Probability: P(6) = 1/6 (1 out of 6).
  • Odds in favour of rolling a 6: 1 : 5 (read "1 to 5").
  • Odds against rolling a 6: 5 : 1.

In general, using A′ for "not A":

odds in favour of A = P(A) : P(A′) and odds against A = P(A′) : P(A).

Odds are written as ratios in lowest terms, usually with whole numbers. Odds are common in sports reports, betting and some medical studies, so you need to be able to move between odds and probability.

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A bag holds 3 maple-leaf tokens and 7 snowflake tokens. P(maple leaf) = 3/10, but the odds in favour of drawing a maple leaf are 3 : 7 (favourable : unfavourable).

From odds back to probability

If the odds in favour of A are a : b, think of a + b equal parts: a of them are A and b are not. So

P(A) = a / (a + b).

If the odds against A are a : b, then the odds in favour are b : a, and P(A) = b/(a + b). Read carefully which kind of odds you are given.

A happensA does not happen3 parts5 partsOdds in favour of A = 3 : 5Odds against A = 5 : 3P(A) = 3/(3 + 5) = 3/8P(not A) = 5/8
Odds of 3 : 5 in favour split the possibilities into 3 + 5 = 8 equal parts. A takes 3 of them, so P(A) = 3/8.
Worked example

Example 1: Odds from counting

One card is drawn from a standard deck. Find the odds in favour of and against drawing (a) a heart, (b) a face card.

  1. (a) Hearts: 13 favourable; not hearts: 39 unfavourable.
  2. Odds in favour = 13 : 39 = 1 : 3. Odds against = 3 : 1.
  3. (b) Face cards: 12 favourable; 40 unfavourable.
  4. Odds in favour = 12 : 40 = 3 : 10. Odds against = 10 : 3.

(a) In favour 1 : 3, against 3 : 1. (b) In favour 3 : 10, against 10 : 3.

Worked example

Example 2: Probability to odds

The probability of snow in Timmins tomorrow is forecast as 0.65. Find the odds in favour of snow.

  1. P(snow) = 0.65, so P(no snow) = 1 − 0.65 = 0.35.
  2. Odds in favour = 0.65 : 0.35.
  3. Multiply both terms by 100 and simplify: 65 : 35 = 13 : 7.

Odds in favour of snow are 13 : 7.

Worked example

Example 3: Odds to probability

(a) The odds in favour of a team winning are 5 : 2. Find P(win).

(b) The odds against a student being chosen for a draw are 24 : 1. Find P(chosen).

  1. (a) a = 5, b = 2. P(win) = 5/(5 + 2) = 5/7 ≈ 0.714.
  2. (b) Odds against 24 : 1 means odds in favour 1 : 24.
  3. P(chosen) = 1/(1 + 24) = 1/25 = 0.04.

(a) 5/7, (b) 1/25.

ProbabilityP(A) = a/(a + b)Odds in favoura : bP(A) : P(not A)add the partsExample: P(A) = 0.2 → 0.2 : 0.8 = 1 : 4Example: odds 7 : 2 → P(A) = 7/9
Moving between probability and odds. To get odds, compare P(A) with P(not A). To get a probability from odds a : b in favour, divide a by the total number of parts a + b.

Payout odds and fair games

In games of chance, the payout odds tell you how much you win. "Pays 4 to 1" means a $1 bet that wins returns your $1 plus $4 profit.

A game is fair when the payout odds equal the true odds against winning. If you roll a die and win on a 6, the true odds against winning are 5 : 1. A payout of 5 to 1 would be fair: over 6 rolls, on average you lose $1 five times and win $5 once, so you break even.

Casinos and lotteries pay less than the true odds on almost every bet. That gap is how the operator makes money on average, and it is why, in the long run, players as a group lose. You will measure this exactly with expected value in the unit Discrete Probability Distributions.

Worked example

Example 4: Is the game fair?

A fundraiser game: you pay $1 and draw a card from a standard deck. If it is an ace, you get your $1 back plus $10. Find the true odds against drawing an ace and decide whether the game is fair.

  1. Aces: 4; not aces: 48. True odds against = 48 : 4 = 12 : 1.
  2. The game pays 10 to 1, which is less than 12 to 1.
  3. Check with 13 plays (the deck ratio): on average 1 win (+$10) and 12 losses (−$12), a net loss of $2.

The game is not fair: it pays 10 to 1 while the true odds against are 12 to 1. On average the fundraiser keeps about $2 for every 13 plays, which is fine for a fundraiser, as long as players know it.

Odds are not probabilities

"The odds are 1 : 4" means P = 1/5, not 1/4. A quick check: the two numbers in odds add up to the total, so the denominator of the probability is a + b. Also, media often say "odds" when they mean probability ("the odds of being struck by lightning are 1 in 1 million"). "1 in n" is a probability; "1 to n" is odds.

Just for fun

Sign spotted on a math teacher's desk: "The odds that I'll accept late homework are 1 : 0." Wait, read that again. 1 : 0 means P = 1/(1 + 0) = 1, so late homework is always accepted. Somebody needs this lesson.

Hands-on activity

Make a fair game

You need: a deck of cards, coins or tokens to keep score (no real money needed), and a family member to play.

  1. Design a simple game: a player wins if they draw a card of a certain type (for example "a club", "a red face card", "a 7 or an 8"). Choose one.
  2. Find the true odds against winning.
  3. Set the payout so the game is fair. Then set a second payout that is not fair (in favour of the "house").
  4. Play 26 rounds with each payout, shuffling and replacing the card each time, and keep a running score in tokens.
  5. Who came out ahead each time? Did the fair game end near 0?

What to look for: for "a club", the true odds against are 39 : 13 = 3 : 1, so a fair payout is 3 to 1. With 26 rounds the fair game should end fairly close to 0, but chance can push it either way; the unfair game should usually favour the house. More rounds make the pattern clearer.

Your turn

1

A die is rolled. Find the odds in favour of rolling a number greater than 4.

2

Find the odds against drawing a red card from a standard deck.

3

P(A) = 0.15. Find the odds in favour of A.

4

The odds in favour of rain are 2 : 3. Find P(rain).

5

The odds against a horse winning are 9 : 2. Find P(the horse wins).

6

Two dice are rolled. Find the odds in favour of a sum of 7.

7

A game pays 3 to 1 if you toss two heads with two coins. Is it fair?

More practice: warm-up

1

Odds in favour are 1 : 4. What are the odds against?

2

Odds in favour are 1 : 4. What is the probability?

3

P(A) = 1/3. Odds in favour?

4

P(A) = 1/2. Odds in favour?

5

Is "1 in 8" a probability or odds?

More practice: on your own

1

A bag has 6 red, 4 blue and 2 white marbles. Find the odds against drawing a blue marble.

2

A sports website gives the odds in favour of a team winning as 7 : 13. Find P(the team does not win).

3

P(A) = 0.08. Find the odds against A.

4

A raffle sells 500 tickets and draws one winner. You buy 4 tickets. Find your odds in favour of winning.

5

Explain why the odds in favour of A and the odds in favour of A′ are reciprocals of each other.

Challenge

1

The odds in favour of A are a : b. Show that P(A′) = b/(a + b), and find the odds in favour of A if P(A) = P(A′)/4.

2

A roulette wheel has 38 equally likely pockets. A bet on a single number pays 35 to 1. Find the true odds against winning and explain who benefits.

3

The odds in favour of event A are 2 : 3, and B is an event with P(B) = P(A) + 0.1. Find the odds against B.

Check your understanding

Quiz

1.

A die is rolled. What are the odds in favour of rolling a 1 or a 2?

2.

The odds in favour of an event are 3 : 7. What is its probability?

3.

The odds against an event are 11 : 4. What is its probability?

4.

P(A) = 0.2. What are the odds against A?

5.

A game pays 4 to 1 when you draw a heart from a standard deck. Which statement is true?

6.

A news story says, "The odds of winning are 1 in 20." What is the probability of winning?

For parents and teachers

Big idea: odds compare favourable with unfavourable outcomes (a : b), while probability compares favourable with all outcomes (a/(a + b)). The 2007 MDM4U expectations do not use the word "odds", but it appears in most MDM4U textbooks and in everyday life, and it is a direct application of theoretical probability (A1.3) and of probability as a measure of likelihood (A1.1).

How to help: Use sports or weather. "If the odds are 3 to 1 against our team, what is the probability they win?" (1/4). The fair-game card activity is a good, money-free way to show why casinos and lotteries always come out ahead.

Common difficulties:

  • Writing odds 1 : 4 as probability 1/4 (it is 1/5).
  • Mixing up odds in favour and odds against.
  • Confusing "1 in n" (probability) with "1 to n" (odds).
  • Forgetting to reduce odds to lowest terms.