Question about binomial distribution and permutation.

Question

A multiple-choice test has 15 questions, each having 4 possible answers, of which only 1 is correct. If the questions are answered at random, what is the probability of getting all of them right?

I tried doing using binomial distributions only: $$ \binom{15}{15} \cdot \left(\frac14\right)^{15} \cdot \left(\frac34\right)^{0} $$ but I don't understand about the "answered at random" part as I am not sure of what to do.

Thanks!!

Answer 1

Questions answered at random means you randomly (uniformly) pick an answer to every question. If so,

1. What is $p$, the probability of getting one specific question correctly?
2. So what is $q$, the probability of not getting one specific question correctly?
3. Now you run $15$ such tests, where you need to get all of them wrong, what it is the probability of that?
4. Finally, what is the probability of getting all of them right?

These do not require the use of binomial variables, but do it from first principles instead. If you like, you can do it using binomial variables as well.