Pick a card from 52, P(face card or heart) 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 a face card or a heart?

1 Find each one

Call face card A.

There are 12.

A = face card → 12 cards

P stands for probability.

There are 52 outcomes in all.

P = probability, all = 52

Call heart B.

There are 13.

B = heart → 13 cards

What is in both?

A and B = 3 cards

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: 12/52, plus 13/52, minus 3/52.

P(A or B) = 1252 + 1352 − 352

Careful!

That counts the middle twice.

Careful: not 1252 + 1352 = 2552 ✗

Same bottoms: we add and take away the tops.

P(A or B) = 12 + 13 − 352

12 plus 13 minus 3 is 22.

So it is 22/52.

P(A or B) = 2252

4 Simplify

Both 22 and 52 can be divided by 2.

That makes it simpler.

P(A or B) = 22 ÷ 252 ÷ 2

So it is 11/26.

P(A or B) = 1126

5 Check

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

9 plus 3 plus 10 is 22.

22 out of 52.

The same.

So the answer is right!

in the circles: 9 + 3 + 10 = 22 → 2252

Let's look back.

First: find each one.

Then: draw a Venn diagram.

Then: add, take away the overlap.

Then: simplify.

Last: check.

Answer