Worked examples
The question
Question: A fair die is rolled.
What is the probability of getting a 1 or a 6?
1 Find each one
Call 1 A.
That is 1.
A = 1 → 1
P stands for probability.
There are 6 outcomes in all.
P = probability, all = 6
Call 6 B.
That is 6.
B = 6 → 6
Is anything in both?
No.
Nothing is A and B at the same time.
A and B = nothing
2 Draw a Venn diagram
3 Add them
A or B: we add them.
Nothing is counted twice.
P(A or B) = P(A) + P(B)
When two things can't happen together, we call them mutually exclusive.
mutually exclusive:
they can't happen together
In numbers: 1/6, plus 1/6.
P(A or B) = 16 + 16
Careful!
Nothing is in both, so there is nothing to take away.
Careful: nothing in both: nothing to take away
Same bottoms: we add and take away the tops.
P(A or B) = 1 + 16
1 plus 1 is 2.
So it is 2/6.
P(A or B) = 26
4 Simplify
Both 2 and 6 can be divided by 2.
That makes it simpler.
P(A or B) = 2 ÷ 26 ÷ 2
So it is 1/3.
P(A or B) = 13
5 Check
Let's check: we count what is inside the circles.
1 and 6: 2.
2 out of 6.
The same.
So the answer is right!
in the circles: 1, 6 → 26
Let's look back.
First: find each one.
Then: draw a Venn diagram.
Then: add them.
Then: simplify.
Last: check.
▶ Watch it solved
The question
Question: A fair die is rolled.
What is the probability of getting an even number or a number more than 4?
1 Find each one
Call even A.
That is 2, 4 and 6.
A = even → 2, 4, 6
P stands for probability.
There are 6 outcomes in all.
P = probability, all = 6
Call more than 4 B.
That is 5 and 6.
B = more than 4 → 5, 6
What is in both?
6.
A and B = 6
2 Draw a Venn diagram
3 Add, take away the overlap
A or B: we add them.
But the middle is in A and in B, so it was counted twice.
We take it away once.
P(A or B) = P(A) + P(B) − P(A and B)
or: add, then take away
the overlap once
In numbers: 3/6, plus 2/6, minus 1/6.
P(A or B) = 36 + 26 − 16
Careful!
That counts the middle twice.
Careful: not 36 + 26 = 56 ✗
Same bottoms: we add and take away the tops.
P(A or B) = 3 + 2 − 16
3 plus 2 minus 1 is 4.
So it is 4/6.
P(A or B) = 46
4 Simplify
Both 4 and 6 can be divided by 2.
That makes it simpler.
P(A or B) = 4 ÷ 26 ÷ 2
So it is 2/3.
P(A or B) = 23
5 Check
Let's check: we count what is inside the circles.
2, 4, 5 and 6: 4.
4 out of 6.
The same.
So the answer is right!
in the circles: 2, 4, 5, 6 → 46
Let's look back.
First: find each one.
Then: draw a Venn diagram.
Then: add, take away the overlap.
Then: simplify.
Last: check.
▶ Watch it solved
The question
Question: A number from 1 to 20 is picked at random.
What is the probability it is even or a multiple of 5?
1 Find each one
Call even A.
That is 10 numbers.
A = even → 10 numbers
P stands for probability.
There are 20 outcomes in all.
P = probability, all = 20
Call multiple of 5 B.
That is 5, 10, 15 and 20.
B = multiple of 5 → 5, 10, 15, 20
What is in both?
10 and 20.
A and B = 10, 20
2 Draw a Venn diagram
3 Add, take away the overlap
A or B: we add them.
But the middle is in A and in B, so it was counted twice.
We take it away once.
P(A or B) = P(A) + P(B) − P(A and B)
or: add, then take away
the overlap once
In numbers: 10/20, plus 4/20, minus 2/20.
P(A or B) = 1020 + 420 − 220
Careful!
That counts the middle twice.
Careful: not 1020 + 420 = 1420 ✗
Same bottoms: we add and take away the tops.
P(A or B) = 10 + 4 − 220
10 plus 4 minus 2 is 12.
So it is 12/20.
P(A or B) = 1220
4 Simplify
Both 12 and 20 can be divided by 4.
That makes it simpler.
P(A or B) = 12 ÷ 420 ÷ 4
So it is 3/5.
P(A or B) = 35
5 Check
Let's check: we count what is inside the circles.
There are 12.
12 out of 20.
The same.
So the answer is right!
in the circles: 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20 → 1220
Let's look back.
First: find each one.
Then: draw a Venn diagram.
Then: add, take away the overlap.
Then: simplify.
Last: check.
▶ Watch it solved
