Step by step
The question
Question: A fair die is rolled.
What is the probability of getting an even number or a prime number?
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 prime B.
That is 2, 3 and 5.
B = prime → 2, 3, 5
What is in both?
2.
A and B = 2
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 3/6, minus 1/6.
P(A or B) = 36 + 36 − 16
Careful!
That counts the middle twice.
Careful: not 36 + 36 = 66 ✗
Same bottoms: we add and take away the tops.
P(A or B) = 3 + 3 − 16
3 plus 3 minus 1 is 5.
So it is 5/6.
P(A or B) = 56
4 Check
Let's check: we count what is inside the circles.
2, 3, 4, 5 and 6: 5.
5 out of 6.
The same.
So the answer is right!
in the circles: 2, 3, 4, 5, 6 → 56
Let's look back.
First: find each one.
Then: draw a Venn diagram.
Then: add, take away the overlap.
Last: check.
Answer
P(A or B) = 56