Step by step
The question
Question: A quiz has 6 questions with 2 choices each.
Every answer is guessed.
What is the probability of exactly 5 right?
1 Binomial: yes or no
6 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: 6.
k is how many right we want: 5.
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
6 C 5 counts the orders: which 5 of the 6 questions are the right.
6 times 5 times 4 times 3 times 2, over 5 times 4 times 3 times 2 times 1, is 6.
Here they are: R is a right answer, W is a wrong answer.
6 orders.
4 One order's chance
One order, like RRRRRW.
Along one path we multiply: 1/2 for each right answer, and 1/2 for each wrong answer.
Top times top, bottom times bottom: 1/64.
5 Multiply
Every order has this same chance.
There are 6 orders, so we multiply by 6.
Careful!
That is only one order.
Right can come in 6 different orders.
6 times 1 is 6.
So it is 6/64.
Both 6 and 64 can be divided by 2.
That makes it simpler.
So it is 3/32.
6 Check
Let's check another order, WRRRRR.
The same number of each, just in other places.
Multiplied, it is 1/64 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.