Consider the Monty Hall problem:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

(You have observed that week after week on this show, the host always opens a door to revival a goat before giving you the option to switch)

1.  Without using conditional probabilities work out the probability of winning if you switch versus if you don’t.
2.  Compute the same probabilities using conditional probabilities.
3. Now assume that the host is cruel. He only offers you a switch when you have chosen the car. (You figure this out by watching the show on TV before going.) Now should you switch when offered the chance ?
4. Now lets assume that every morning the host gets up and with probability \(p\) acts like the original host and with probability \(1-p\) he acts like the cruel host from the previous question.
   1. Now what is the chance  of wining if you switch ?
   2. Now what is the chance  of wining if you don’t switch ?
   3. For which \(p\) should you switch and which \(p\) should you not switch ?
5. Now let us assume that host just picks a door randomly after you have picked yours. If there is a car behind it the came is over. If there is a goat, he offers you the chance to switch. Should you switch when you have the chance ? why ?