Worked examples
The question
Question: A fair die is rolled.
What are the odds for getting a number more than 4?
The odds for something compare the outcomes for it, with the outcomes against it.
odds for = for : against
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: numbers more than 4.
2 outcomes are for.
for = 2
All the others are against: 6 minus 2 is 4.
against = 6 − 2 = 4
3 Write the odds
For, then against: 2 to 4.
odds for = 2 : 4
Careful!
Odds compare for with against.
Not with all 6.
2/6 would be a probability.
Careful: not 2 : 6 ✗
4 Simplify
Like a ratio: both 2 and 4 can be divided by 2.
odds for = (2 ÷ 2) : (4 ÷ 2)
So the odds are 1 to 2.
odds for = 1 : 2
5 Check
Let's check: for and against together are 2 plus 4, that is 6.
Every outcome is counted once.
So the answer is right!
for + against = 2 + 4 = 6 = all
Let's look back.
First: count the outcomes.
Then: count for and against.
Then: write the odds.
Then: simplify.
Last: check.
▶ Watch it solved
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 ✗
▶ Watch it solved
The question
Question: Two dice are rolled.
What are the odds against the sum being 7?
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
The first die goes across.
The second die goes down.
Each box is one outcome.
We write the sum of the two dice in it.
So there are 36 outcomes in all, each just as likely.
all = 36
2 Count for and against
The outcomes we want: sums of 7.
6 outcomes are for.
for = 6
All the others are against: 36 minus 6 is 30.
against = 36 − 6 = 30
3 Write the odds
Against, then for: 30 to 6.
odds against = 30 : 6
Careful!
Odds compare against with for.
Not with all 36.
30/36 would be a probability.
Careful: not 30 : 36 ✗
4 Simplify
Like a ratio: both 30 and 6 can be divided by 6.
odds against = (30 ÷ 6) : (6 ÷ 6)
So the odds are 5 to 1.
odds against = 5 : 1
5 Check
Let's check: for and against together are 6 plus 30, that is 36.
Every outcome is counted once.
So the answer is right!
for + against = 6 + 30 = 36 = all
Let's look back.
First: count the outcomes.
Then: count for and against.
Then: write the odds.
Then: simplify.
Last: check.
Answer odds against = 5 : 1
▶ Watch it solved
