Question

On a multiple choice test with 20 questions, each question has four possible answers, one of which is correct. If 10 students sit for the test, what is the probability that one student does not get any correct answer at all?

I'm stuck at b :(

Answer 1

`~~3.11%`

Explanation:

We first need to assume that all of the students are guessing on all the questions (this would have to be the worst test ever given to a class...)

The probability of achieving no correct answers can be found by taking the probability of getting one answer wrong, which is `3/4`, and taking that to the 20th power:

`(3/4)^20~~0.0032`

We can find the probability of exactly 1 student not getting any questions right by using the binomial probability, the general formula of which is:

`sum_(k=0)^(n)C_(n,k)(p)^k(~p)^(n-k)`

We're going to set `p=0.0032`, making `~p=0.9968`. `n=10, k=1`:

`C_(10,1)(.0032)^1(.9968)^(9)~~0.0311`

And so, roughly speaking, there is a slightly better than 3% chance that one student will get all the questions wrong.