Step by step
The question
Question: 5 cards are dealt from a deck of 52.
What is the probability that all 5 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 5: the top 5 numbers from 52 down, over 5 factorial.
Before we multiply, we make the numbers smaller.
50 and 5 both divide by 5: 10.
52 and 4 both divide by 4: 13.
51 and 3 both divide by 3: 17.
10 and 2 both divide by 2: 5.
What is left is 13 times 17 times 5 times 49 times 48.
13 times 17 is 221.
221 times 5 is 1105.
1105 times 49 is 54,145.
54,145 times 48 is 2,598,960.
So there are 2,598,960 ways.
3 Count the ways we want
13 C 5: the top 5 numbers from 13 down, over 5 factorial.
Before we multiply, we make the numbers smaller.
10 and 5 both divide by 5: 2.
12 and 4 both divide by 4: 3.
3 and 3 both divide by 3: 1.
2 and 2 both divide by 2: 1.
What is left is 13 times 11 times 9.
13 times 11 is 143.
143 times 9 is 1287.
So there are 1287 ways.
4 Write the fraction
1287 ways we want, out of 2,598,960.
Careful!
The top and the bottom must count the same way: both are groups, with C.
The numbers are big, so we use a decimal: 1287 divided by 2,598,960 is about 0.000495.
Very small!
5 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 about 0.000495.
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.
Last: check.