Pick a card from 52, P(ace or spade) Step by Step

Step by step

The question

Question: A card is picked from a deck of 52 cards.

What is the probability it is an ace or a spade?

1 Find each one

Call ace A.

There are 4.

A = ace → 4 cards

P stands for probability.

There are 52 outcomes in all.

P = probability, all = 52

Call spade B.

There are 13.

B = spade → 13 cards

What is in both?

The ace of spades: 1 card.

A and B = ace of spades → 1 card

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: 4/52, plus 13/52, minus 1/52.

P(A or B) = 452 + 1352 − 152

Careful!

That counts the middle twice.

Careful: not 452 + 1352 = 1752 ✗

Same bottoms: we add and take away the tops.

P(A or B) = 4 + 13 − 152

4 plus 13 minus 1 is 16.

So it is 16/52.

P(A or B) = 1652

4 Simplify

Both 16 and 52 can be divided by 4.

That makes it simpler.

P(A or B) = 16 ÷ 452 ÷ 4

So it is 4/13.

P(A or B) = 413

5 Check

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

3 plus 1 plus 12 is 16.

16 out of 52.

The same.

So the answer is right!

in the circles: 3 + 1 + 12 = 16 → 1652

Let's look back.

First: find each one.

Then: draw a Venn diagram.

Then: add, take away the overlap.

Then: simplify.

Last: check.

Answer