One compares the two sides to each other. The other compares one side to everybody.
The problem
The odds in favour of rain are 3 : 7. What is the probability of rain?
Odds compare favourable to unfavourable, so the whole is 3 + 7 = 10 and P(rain) = 310. B is the odds ratio written as a fraction — odds and probability have different denominators, and that is the entire lesson.
wins : losses
3 : 7 — part against part
wins ÷ everything
3 out of 10 — part of the whole
The 7 counts only the failures, so it is not the total. Reading odds straight across as a fraction is the mistake this whole topic exists to catch — and 37 will be one of the choices.
P = favourablefavourable + unfavourable
3 + 7 = 10
the whole crowd
P(rain) = 310
0.3, or 30%
P(no rain) = 710
the other part
Everyday speech nearly always quotes odds against — “the odds are ten to one” means one win for every ten losses, a probability of 111. Maths questions say which they mean, so read the word before the colon.
A bag has 4 red marbles out of 10. Give the odds in favour of red and the odds against. The numerator is one part; what is left over is the other.
P(red) = 410 = 25
4 red out of 10 marbles
not red = 10 − 4 = 6
the marbles that are not red
odds in favour = 4 : 6 = 2 : 3
simplify like any ratio
odds against = 3 : 2
flip the ratio: not red to red
| Probability | Odds in favour | Odds against |
|---|---|---|
| 12 | 1 : 1 — even | 1 : 1 |
| 14 | 1 : 3 | 3 : 1 |
| 25 | 2 : 3 | 3 : 2 |
| 34 | 3 : 1 | 1 : 3 |
| 16 | 1 : 5 | 5 : 1 |
Notice the pattern: the two odds numbers always add up to the probability's denominator, once the fraction is in lowest terms.
Odds of 7 : 3 are “more than one”, which is perfectly normal. Any probability above 1 or below 0 is a sign you divided by the wrong total — usually the losing part instead of the whole.
A decimal probability. The probability of an event is 0.15. What are the odds in favour?
0.15 = 15100
think of 100 equal outcomes
15 for, 100 − 15 = 85 against
split the whole
15 : 85 = 3 : 17
divide both by 5
What is different: turn the decimal into a fraction first, then split its denominator. Check: 3 + 17 = 20 and 320 = 0.15 ✓.
Odds from a sample space. Two dice are rolled. What are the odds against a sum of 9?
sum 9: (3,6), (4,5), (5,4), (6,3) — 4 ways
out of 36
not 9: 36 − 4 = 32 ways
the losing part
against = 32 : 4 = 8 : 1
losses first, then simplify
What is different: nothing is handed over as a ratio. Count the favourable and unfavourable outcomes, then write them in the order the question asks.
36 : 4 = 9 : 1 compares the whole with the wins. Odds compare losses with wins, so the 4 winning rolls come out of the 36 first.
white = k, black = 3k
odds 1 : 3 in parts
after: k + 6 white, 3k black
only the white count changes
k + 6 = 3k
new odds 1 : 1 means equal counts
k = 3
at first 3 + 9 = 12 chips
Check: after adding there are 9 white and 9 black — odds 1 : 1 ✓.
1 + 6 : 3 is not how ratios grow. Turn the odds into actual counts with one unknown k first.
Odds items switch between odds and probability (add the parts to get the total), swap “in favour” and “against”, or give the odds with one actual count and ask for the rest. Always check which side is listed first. Ten-second check: probability = favourable part ÷ sum of the parts, and it must be below 1.
The problem
The odds against a team winning are 5 : 4. Find the probability that it wins.
"Against" puts the unfavourable side first, so it is 4 winning to 5 losing, out of 9. P(win) = 49. C is the probability of losing — the single word "against" reverses the order.
