1. For ?xed B such that P(B) > 0, show that P(·|B) satis?es the axioms of probability.
2. You have probably heard it before. Now you can solve it rigorously. It is called the "Monty Hall Problem." A prize is placed at random behind one of three doors. You pick a door. To be concrete, let's suppose you always pick door 1. Now Monty Hall chooses one of the other two doors, opens it and shows you that it is empty. He then gives you the opportunity to keep your door or switch to the other unopened door. Should you stay or switch? Intuition suggests it doesn't matter. The correct answer is that you should switch. Prove it. It will help to specify the sample space and the relevant events carefully. Thus write ? = {(ω1, ω2) : ωi ∈ {1, 2, 3}} where ω1 is where the prize is and ω2 is the door Monty opens.