What is relationship between beta and binomial distributions in Bayesian inference I came across this question:
Suppose we are giving two students a multiple-choice exam with 40 questions, where each question has four choices. We don't know how much the students have studied for this exam, but we think that they will do better than just guessing randomly. a: What is our likelihood? b: What prior should we use?
The solutions were: a: Likelihood is Binomial(40, theta); b: The conjugate prior is a beta prior.
Could someone please explain why beta is the conjugate prior of a binomial? I meant how could one know $\theta$ was distributed as $Beta$? Could other distributions be used for binomial likelihood, and what is the consequence of not using $Beta$ for the prior?
Thanks in advance.

The point is that if the prior is a beta distribution and the likelihood comes from a binomial distribution, then the posterior is a again a beta distribution.
