Question

Aaron, Ben and Celia are runing a race. Assume that all possible rankings at the finishing line are equally likely. Ben always wears a green bib, Celia always wears an orange bib and Aaron wears a green bib with probability p and an orange bib with probability 1 − p, p ∈ (0, 1). If the winner of the race was in an orange bib and the second person to finish the race was in a green bib, compute now the probabilities of all possible rankings at the finishing line.

Answer 1

P[ Aaron wins ] = 1/3

P[ Ben wins ] = 1/3

P[ Celia wins ] = 1/3

P[ Aaron wears an orange bib ] = 1-p

P[ Aaron wears an green bib ] = p

P[ Celia wears an orange bib ] = 1

P[ Ben wears an green bib ] = 1

Winner wearing orange bib and second person wearing green bib,

Possible rankings are: {C,B,A}, {C,A,B}, {A,B,C}

P[ C,B,A ] = P[ Celia wins ] = 1/3

P[ C,A,B ] = P[ Celia wins ]*P[ Aaron wearing green bib ] = 1/3*(1-p) = (1-p)/.3

P[ A,B,C ] = P[ Aaron wins ]*P[ Aaron wearing an orange bib ] = 1/3*p = p/3