Pick a number from 1 to 20, P(even or multiple of 5) Step by Step

Step by step

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.

Answer