Pick a card from 52, P(jack or black) 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 jack or black?

1 Find each one

Call jack A.

There are 4.

A = jack → 4 cards

P stands for probability.

There are 52 outcomes in all.

P = probability, all = 52

Call black B.

There are 26.

B = black → 26 cards

What is in both?

A and B = 2 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: 4/52, plus 26/52, minus 2/52.

P(A or B) = 452 + 2652 − 252

Careful!

That counts the middle twice.

Careful: not 452 + 2652 = 3052 ✗

Same bottoms: we add and take away the tops.

P(A or B) = 4 + 26 − 252

4 plus 26 minus 2 is 28.

So it is 28/52.

P(A or B) = 2852

4 Simplify

Both 28 and 52 can be divided by 4.

That makes it simpler.

P(A or B) = 28 ÷ 452 ÷ 4

So it is 7/13.

P(A or B) = 713

5 Check

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

2 plus 2 plus 24 is 28.

28 out of 52.

The same.

So the answer is right!

in the circles: 2 + 2 + 24 = 28 → 2852

Let's look back.

First: find each one.

Then: draw a Venn diagram.

Then: add, take away the overlap.

Then: simplify.

Last: check.

Answer