Step by step
The question
Question: A number from 1 to 15 is picked at random.
What is the probability it is a multiple of 3 or a multiple of 5?
1 Find each one
Call multiple of 3 A.
That is 3, 6, 9, 12 and 15.
A = multiple of 3 → 3, 6, 9, 12, 15
P stands for probability.
There are 15 outcomes in all.
P = probability, all = 15
Call multiple of 5 B.
That is 5, 10 and 15.
B = multiple of 5 → 5, 10, 15
What is in both?
15.
A and B = 15
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: 5/15, plus 3/15, minus 1/15.
P(A or B) = 515 + 315 − 115
Careful!
That counts the middle twice.
Careful: not 515 + 315 = 815 ✗
Same bottoms: we add and take away the tops.
P(A or B) = 5 + 3 − 115
5 plus 3 minus 1 is 7.
So it is 7/15.
P(A or B) = 715
4 Check
Let's check: we count what is inside the circles.
There are 7.
7 out of 15.
The same.
So the answer is right!
in the circles: 3, 5, 6, 9, 10, 12, 15 → 715
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) = 715