Step by step
The question
Question: A quiz has 5 questions with 4 choices each.
Every answer is guessed.
What is the probability of exactly 2 right?
1 Binomial: yes or no
5 questions.
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 questions: 5.
k is how many right we want: 2.
p is the chance of a success each time: 1/4.
q is the chance of not: 1 minus p, 3/4.
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 2 counts the orders: which 2 of the 5 questions are the right.
5 times 4, over 2 times 1, is 10.
Here they are: R is a right answer, W is a wrong answer.
10 orders.
4 One order's chance
One order, like RRWWW.
Along one path we multiply: 1/4 for each right answer, and 3/4 for each wrong answer.
Top times top, bottom times bottom: 27/1024.
5 Multiply
Every order has this same chance.
There are 10 orders, so we multiply by 10.
Careful!
That is only one order.
Right can come in 10 different orders.
10 times 27 is 270.
So it is 270/1024.
Both 270 and 1024 can be divided by 2.
That makes it simpler.
So it is 135/512.
As a decimal, to 4 places, that is about 0.2637.
6 Check
Let's check another order, WWWRR.
The same number of each, just in other places.
Multiplied, it is 27/1024 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.