3.2 Exercise 21

An individual is to select from among n alternatives in an attempt to obtain a particular one. This might be selection from answers on a multiple choice question, when only one is correct. Let A be the event he makes a correct selection, and B be the event he knows which is correct before making the selection. We suppose $P \left(B\right) = p$ and $P \left(\right. A \left|\right. B^{c} \left.\right) = 1 / n$. Determine $P \left(\right. B \left|\right. A \left.\right)$; show that $P \left(\right. B \left|\right. A \left.\right) \geq P \left(B\right)$ and $P \left(\right. B \left|\right. A \left.\right)$ increases with n for fixed p.