Monty Hall Problem

This is a very famous probability problem. This problem illustrates that probability is sometimes more convincing than "what we perceive as obvious". I'll post another problem that defeats "obvious reasoning".

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, 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?

Solution: show

Well for a mathematician stuck on the game show, I would suggest tossing the unbiased coin. The reason being that Game Show host might be clever and seeing that mathematician got the prize behind door 1, would play this card in order to entice the mathematician to switch. Mathematician can be well aware of this trick of Game show host and might not switch. But game show host thought so .. errr. This is a recursive logic and would go to infinity.

Under the assumption that game show host has no hidden motive and performs this step always, the best choice is to switch. That increases the chances by 33.33%. There is a very simple explanation to this. The prize could be behind any doors. Since you pick door 1. Assume game show hosts says that either keep door 1 or take both door 2 and 3, you would go for both door. That doubles up wining chances to 66%. He opens door 3 for you and you open door 2 when you say switch.

Red Green Cards: A box has three cards. First card has both sides green, second card has both sides red, third card has one side green and one side red. A card is picked at random and its one side is observed to be green. What is the probability that the other side of the card is also green.

Solution: show

Well it seems that answer should be 1/2. But its 2/3. For explanation, let us label card 1 as g1|g2, second card as r1|r2, third card as g3|r3. Now since one side of the card is green. It can either be g1, g2, g3. Three events and two point to first card.