Step by step
The question
Question: A quiz has 4 questions with 3 choices each.
Every answer is guessed.
What is the probability of exactly 1 right?
1 Binomial: yes or no
4 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: 4.
k is how many right we want: 1.
p is the chance of a success each time: 1/3.
q is the chance of not: 1 minus p, 2/3.
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 1 counts the orders: the one right answer can be any of the 4 questions.
That is 4.
Here they are: R is a right answer, W is a wrong answer.
4 orders.
4 One order's chance
One order, like RWWW.
Along one path we multiply: 1/3 for each right answer, and 2/3 for each wrong answer.
Top times top, bottom times bottom: 8/81.
5 Multiply
Every order has this same chance.
There are 4 orders, so we multiply by 4.
Careful!
That is only one order.
Right can come in 4 different orders.
4 times 8 is 32.
So it is 32/81.
6 Check
Let's check another order, WWWR.
The same number of each, just in other places.
Multiplied, it is 8/81 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.