Step by step
The question
Question: A number from 1 to 10 is picked at random.
What is the probability it is even or a multiple of 3?
1 Find each one
Call even A.
That is 2, 4, 6, 8 and 10.
A = even → 2, 4, 6, 8, 10
P stands for probability.
There are 10 outcomes in all.
P = probability, all = 10
Call multiple of 3 B.
That is 3, 6 and 9.
B = multiple of 3 → 3, 6, 9
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: 5/10, plus 3/10, minus 1/10.
P(A or B) = 510 + 310 − 110
Careful!
That counts the middle twice.
Careful: not 510 + 310 = 810 ✗
Same bottoms: we add and take away the tops.
P(A or B) = 5 + 3 − 110
5 plus 3 minus 1 is 7.
So it is 7/10.
P(A or B) = 710
4 Check
Let's check: we count what is inside the circles.
There are 7.
7 out of 10.
The same.
So the answer is right!
in the circles: 2, 3, 4, 6, 8, 9, 10 → 710
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) = 710