Question

A 10 question multiple choice exam is given and each question has 5 possible answers. A student takes this exam and guesses at every question. What is the probability they get at least 9 questions correct?

Answer 1

`0.0000041984~~4.2xx10^(-6)`

Explanation:

Let's first set up the binomial. The general formula is:

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

We have `n=10`.

With 5 possible answers on each question, this gives the probability of guessing the correct answer `p=1/5`, meaning the probability of getting it wrong is `~p=4/5`.

We're only looking at the probability of getting at least 9 questions correct, and so only care about getting 9 questions correct and 10 questions correct. This gives:

`C_(10,9)(1/5)^9(4/5)^(1)+C_(10,10)(1/5)^10(4/5)^(0)`

`(10)(1/5)^9(4/5)^(1)+(1)(1/5)^10(4/5)^(0)=0.0000041984~~4.2xx10^(-6)` or roughly 3X more likely than being hit by lightening over the course of a year.