Step by step
The question
Question: A fair die is rolled.
What are the odds against getting a 6?
The odds against something put the outcomes against it first, then the outcomes for it.
odds against = against : for
For: the outcomes we want.
Against: all the others.
for = we want, against = the others
1 Count the outcomes
A die has 6 sides: 1, 2, 3, 4, 5 and 6.
So there are 6 outcomes in all, each just as likely.
all = 6
2 Count for and against
The outcomes we want: the 6.
1 outcome is for.
for = 1
All the others are against: 6 minus 1 is 5.
against = 6 − 1 = 5
3 Write the odds
Against, then for: 5 to 1.
odds against = 5 : 1
Careful!
Odds compare against with for.
Not with all 6.
5/6 would be a probability.
Careful: not 5 : 6 ✗
4 Check
Let's check: for and against together are 1 plus 5, that is 6.
Every outcome is counted once.
So the answer is right!
for + against = 1 + 5 = 6 = all
Let's look back.
First: count the outcomes.
Then: count for and against.
Then: write the odds.
Last: check.
Answer
Careful: not 5 : 6 ✗