Step by step
The question
Question: A bag has 3 red, 5 blue and 2 green marbles.
One marble is picked without looking.
What are the odds against it being red?
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
So there are 10 outcomes in all, each just as likely.
all = 10
2 Count for and against
The outcomes we want: red marbles.
3 outcomes are for.
for = 3
All the others are against: 10 minus 3 is 7.
against = 10 − 3 = 7
3 Write the odds
Against, then for: 7 to 3.
odds against = 7 : 3
Careful!
Odds compare against with for.
Not with all 10.
7/10 would be a probability.
Careful: not 7 : 10 ✗
4 Check
Let's check: for and against together are 3 plus 7, that is 10.
Every outcome is counted once.
So the answer is right!
for + against = 3 + 7 = 10 = all
Let's look back.
First: count the outcomes.
Then: count for and against.
Then: write the odds.
Last: check.
Answer
Careful: not 7 : 10 ✗