Step by step
The question
Question: 3 cards are dealt from a deck of 52.
What is the probability that all 3 are kings?
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 3: the top 3 numbers from 52 down, over 3 factorial.
Before we multiply, we make the numbers smaller.
51 and 3 both divide by 3: 17.
52 and 2 both divide by 2: 26.
What is left is 26 times 17 times 50.
26 times 17 is 442.
442 times 50 is 22,100.
So there are 22,100 ways.
3 Count the ways we want
4 C 3: the top 3 numbers from 4 down, over 3 factorial.
Before we multiply, we make the numbers smaller.
3 and 3 both divide by 3: 1.
4 and 2 both divide by 2: 2.
What is left: 2 times 2.
That is 4 ways.
4 Write the fraction
4 ways we want, out of 22,100.
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: 4 divided by 22,100 is about 0.000181.
Very small!
5 Check
Let's check another way: pick one at a time.
The first card: 4 out of 52.
Then one fewer each time.
Multiplied, that is about 0.000181.
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.