> The question as stated does not permit such a choice by the host since if it were a choice, it had already been made. I don't think you understand the concep

> The question as stated does not permit such a choice by the host since if it were a choice, it had already been made. I don't think you understand the concept of conditional probabilities correctly. The fact that event B already happened doesn't make it any easier to compute P(A|B) nor it renders the P(B) useless. On the contrary P(B) and P(AB) are key to solve this problem.