Step by step
The question
Question: 2 cards are dealt from a deck of 52.
What is the probability that both are hearts?
The probability is the number of ways we want, over all the ways.
P stands for probability.
1 Order: yes or no
Does order matter?
No.
A hand of cards is the same hand in any order.
So we count groups: combinations.
2 Count all the ways
52 C 2: the top 2 numbers from 52 down, over 2 factorial.
Before we multiply, we make the numbers smaller.
52 on top and 2 at the bottom: 52 divided by 2 is 26.
What is left: 26 times 51.
That is 1326 ways.
3 Count the ways we want
13 C 2: the top 2 numbers from 13 down, over 2 factorial.
Before we multiply, we make the numbers smaller.
12 on top and 2 at the bottom: 12 divided by 2 is 6.
What is left: 13 times 6.
That is 78 ways.
4 Write the fraction
78 ways we want, out of 1326.
Careful!
The top and the bottom must count the same way: both are groups, with C.
5 Simplify
Both 78 and 1326 can be divided by 78.
That makes it simpler.
So it is 1/17.
6 Check
Let's check another way: pick one at a time.
The first card: 13 out of 52.
Then one fewer each time.
Multiplied, that is 1/17.
The same.
So the answer is right!
Let's look back.
First: order: yes or no.
Then: count all the ways.
Then: count the ways we want.
Then: write the fraction.
Then: simplify.
Last: check.