Question

Problem 1 Three students tA, B, C with seemingly equal records have applied for a major scholarship. They have been notified that two winners have been selected, but not which two. An advisor, who is friends with Student A, knows who will win the scholarships. A realizes it would be unethical to ask if they,A, will win. but thinks of asking for one student's name other than themselves. They believe before asking, their chance of winning is '/s. A think that if they ask and the counsellor says, "B will win," their own chances will go down to ½, because either A and B or B and C will win. And so, A decides not to reduce their chances by asking. However, A is mistaken in their calculations. Explain why and show the correct calculations of the probability before and after asking.

Answer 1

$Let~E_A~denote~that~Student~A~wins~scholarship.~Similarly~for~E_B~and ~E_C.\\ Let~S_A~denote~that~the~counselor~says~that~A~will~win;~similarly~S_B~and~S_C.\\ then~P(E_A^c)=P(E_B^c)=P(E_C^c)=\frac{1}{3}\\ P(S_B|E_A^c)=1/2,~P(S_B|E_B^c)=0,~P(S_B|E_C^c)=1\\ \\ P(E_A|S_B)=1-P(E_A^c|S_B)=1-\frac{P(S_B|E_A^c)P(E_A^c)}{P(S_B)}\\ =1-\frac{P(S_B|E_A^c)P(E_A^c)}{P(S_B|E_A^c)P(E_A^c)+P(S_B|E_B^c)P(E_B^c)+P(S_B|E_C^c)P(E_C^c)}\\ =1-\frac{(1/2)*(1/3)}{(1/2)*(1/3)+0*(1/3)+1*(1/3)}=1-\frac{1}{3}=\frac{1}{3}\\ So~the~posterior~probability~P(E_A|S_B)=~prior~probability~P(E_A)$