Step by step
The question
Question: A die is rolled 4 times.
What is the probability of exactly 2 sixes?
1 Binomial: yes or no
4 rolls.
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 rolls: 4.
k is how many sixes we want: 2.
p is the chance of a success each time: 1/6.
q is the chance of not: 1 minus p, 5/6.
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 rolls are the sixes.
4 times 3, over 2 times 1, is 6.
Here they are: 6 is a six, N is a roll that is not a 6.
6 orders.
4 One order's chance
One order, like 66NN.
Along one path we multiply: 1/6 for each six, and 5/6 for each roll that is not a 6.
Top times top, bottom times bottom: 25/1296.
5 Multiply
Every order has this same chance.
There are 6 orders, so we multiply by 6.
Careful!
That is only one order.
Sixes can come in 6 different orders.
6 times 25 is 150.
So it is 150/1296.
Both 150 and 1296 can be divided by 6.
That makes it simpler.
So it is 25/216.
As a decimal, to 4 places, that is about 0.1157.
6 Check
Let's check another order, NN66.
The same number of each, just in other places.
Multiplied, it is 25/1296 again.
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.