Roll a die, P(odd or less than 3) Step by Step

Step by step

The question

Question: A fair die is rolled.

What is the probability of getting an odd number or a number less than 3?

1 Find each one

Call odd A.

That is 1, 3 and 5.

A = odd → 1, 3, 5

P stands for probability.

There are 6 outcomes in all.

P = probability, all = 6

Call less than 3 B.

That is 1 and 2.

B = less than 3 → 1, 2

What is in both?

A and B = 1

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 2/6, minus 1/6.

P(A or B) = 36 + 26 − 16

Careful!

That counts the middle twice.

Careful: not 36 + 26 = 56 ✗

Same bottoms: we add and take away the tops.

P(A or B) = 3 + 2 − 16

3 plus 2 minus 1 is 4.

So it is 4/6.

P(A or B) = 46

4 Simplify

Both 4 and 6 can be divided by 2.

That makes it simpler.

P(A or B) = 4 ÷ 26 ÷ 2

So it is 2/3.

P(A or B) = 23

5 Check

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

1, 2, 3 and 5: 4.

4 out of 6.

The same.

So the answer is right!

in the circles: 1, 2, 3, 5 → 46

Let's look back.

First: find each one.

Then: draw a Venn diagram.

Then: add, take away the overlap.

Then: simplify.

Last: check.

Answer