Denote by A, B, and C the events that a grand prize is behind doors A, B, and C, respectively. Suppose you randomly picked a door, say A. The game host opened a door, say B, and showed there was no prize behind it. Now the host offers you the option of either staying at the door that you picked (A) or switching to the remaining unopened door (C). Suppose the three doors have equal probability to have the grand prize. When you have picked a door, the host will then open one of the remaining doors that doesn't have the prize with equal probability. Use conditional probability to explain whether you should switch or not. (Hint: Denote by HB the event that the host opened door B. Let A, B and C represent the events that the prize is behind door A, door B, and door C, respectively. Focus on the events where you have opened door A and calculate P(H | A), P(H | B), and P(H | C).)