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?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find each one
Call face card A.
There are 12.
P stands for probability.
There are 52 outcomes in all.
Call heart B.
There are 13.
What is in both?
3.
2 Draw a Venn diagram
The box holds all the outcomes.
A in the blue circle: 9 that are only A, and 3 in the middle, in both.
B in the orange circle: 10 more that are only B.
30 are neither A nor B.
They go outside.
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.
In numbers: 12/52, plus 13/52, minus 3/52.
Careful!
Adding alone counts the middle twice.
Same bottoms: we add and take away the tops.
12 plus 13 minus 3 is 22.
So it is 22/52.
4 Simplify
Both 22 and 52 can be divided by 2.
That makes it simpler.
So it is 11/26.
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!
6 Pick the choice
Our answer is 11/26.
That is choice A.
7 Why not the others
B is 25/52.
Wrong: counted the 3 face hearts twice: take them away once.
That is the Careful mistake from before.
C is 3/52.
Wrong: that is face card and heart.
D is 1/4.
Wrong: that is a heart only.
Let's look back.
First: find each one.
Then: draw a Venn diagram.
Then: add, take away the overlap.
Then: simplify.
Then: check.
Then: pick the choice.
Last: why not the others.