Pick a number from 1 to 12, P(odd or multiple of 3) Step by Step

Step by step

The question

Question: A number from 1 to 12 is picked at random.

What is the probability it is odd or a multiple of 3?

1 Find each one

Call odd A.

That is 1, 3, 5, 7, 9 and 11.

A = odd → 1, 3, 5, 7, 9, 11

P stands for probability.

There are 12 outcomes in all.

P = probability, all = 12

Call multiple of 3 B.

That is 3, 6, 9 and 12.

B = multiple of 3 → 3, 6, 9, 12

What is in both?

3 and 9.

A and B = 3, 9

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: 6/12, plus 4/12, minus 2/12.

P(A or B) = 612 + 412 − 212

Careful!

That counts the middle twice.

Careful: not 612 + 412 = 1012 ✗

Same bottoms: we add and take away the tops.

P(A or B) = 6 + 4 − 212

6 plus 4 minus 2 is 8.

So it is 8/12.

P(A or B) = 812

4 Simplify

Both 8 and 12 can be divided by 4.

That makes it simpler.

P(A or B) = 8 ÷ 412 ÷ 4

So it is 2/3.

P(A or B) = 23

5 Check

Let's check: we count what is inside the circles.

There are 8.

8 out of 12.

The same.

So the answer is right!

in the circles: 1, 3, 5, 6, 7, 9, 11, 12 → 812

Let's look back.

First: find each one.

Then: draw a Venn diagram.

Then: add, take away the overlap.

Then: simplify.

Last: check.

Answer