Step by step
The question
Question: A coin is flipped 4 times.
What is the probability of exactly 2 heads?
1 Binomial: yes or no
4 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: 4.
k is how many heads we want: 2.
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
4 C 2 counts the orders: which 2 of the 4 flips are the heads.
4 times 3, over 2 times 1, is 6.
Here they are: H is a head, T is a tail.
6 orders.
4 One order's chance
One order, like HHTT.
Along one path we multiply: 1/2 for each head, and 1/2 for each tail.
Top times top, bottom times bottom: 1/16.
5 Multiply
Every order has this same chance.
There are 6 orders, so we multiply by 6.
Careful!
That is only one order.
Heads can come in 6 different orders.
6 times 1 is 6.
So it is 6/16.
Both 6 and 16 can be divided by 2.
That makes it simpler.
So it is 3/8.
6 Check
Let's check another order, TTHH.
The same number of each, just in other places.
Multiplied, it is 1/16 again.
And count it: 4 flips make 16 outcomes, all just as likely.
6 of them have exactly 2 heads.
6/16 is 3/8.
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.