Step by step
The question
Question: A coin is flipped 5 times.
What is the probability of exactly 3 heads?
1 Binomial: yes or no
5 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: 5.
k is how many heads we want: 3.
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
5 C 3 counts the orders: which 3 of the 5 flips are the heads.
5 times 4 times 3, over 3 times 2 times 1, is 10.
Here they are: H is a head, T is a tail.
10 orders.
4 One order's chance
One order, like HHHTT.
Along one path we multiply: 1/2 for each head, and 1/2 for each tail.
Top times top, bottom times bottom: 1/32.
5 Multiply
Every order has this same chance.
There are 10 orders, so we multiply by 10.
Careful!
That is only one order.
Heads can come in 10 different orders.
10 times 1 is 10.
So it is 10/32.
Both 10 and 32 can be divided by 2.
That makes it simpler.
So it is 5/16.
6 Check
Let's check another order, TTHHH.
The same number of each, just in other places.
Multiplied, it is 1/32 again.
And count it: 5 flips make 32 outcomes, all just as likely.
10 of them have exactly 3 heads.
10/32 is 5/16.
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.