Step by step
The question
Question: A coin is flipped 6 times.
What is the probability of exactly 4 heads?
1 Binomial: yes or no
6 flips.
Each one has the same chance, and one does not change the next.
Each is a success or not.
That is a binomial experiment.
n is the number of flips: 6.
k is how many heads we want: 4.
p is the chance of a success each time: 1/2.
q is the chance of not: 1 minus p, 1/2.
2 Write the rule
The rule: n C k, times p to the power k, times q to the power n minus k.
We put the numbers in.
3 Count the orders
6 C 4 counts the orders: which 4 of the 6 flips are the heads.
6 times 5 times 4 times 3, over 4 times 3 times 2 times 1, is 15.
4 One order's chance
One order, like this one.
Along one path we multiply: 1/2 for each head, and 1/2 for each tail.
Top times top, bottom times bottom: 1/64.
5 Multiply
Every order has this same chance.
There are 15 orders, so we multiply by 15.
Careful!
That is only one order.
Heads can come in 15 different orders.
15 times 1 is 15.
So it is 15/64.
6 Check
And count it: 6 flips make 64 outcomes, all just as likely.
15 of them have exactly 4 heads.
That is 15/64.
So the answer is right!
Let's look back.
First: binomial: yes or no.
Then: write the rule.
Then: count the orders.
Then: one order's chance.
Then: multiply.
Last: check.