Monday, June 1, 2009

monty hall paradox

Q: 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?

-- table below (somewhere online) lists all the possible scenario on the probability "tree" -- the only effective tree I know. There are many wrong ways to construct the tree. You can see switching = 66% win.

initial pick A A A B B B C C C
prize location A B C A B C A B C
host can open BC C B C AC A B A AB
outcome | switch' W L L L W L L L W
outcome | switch L W W W L W W W L

-----Let me use extremes to illustrate why switching is Always better.

Say we start with 1,000,000,000 doors. Knowing only one of them has the car, our first pick is almost hopeless so we shrug and pick the red wooden oval door. Now the host opens 999,999,998 doors. You are left with your red door and a blue door.

Now we know deep down the red is almost certain to be worthless. Host has helped eliminate 999,999,998 wrong choices, so he is giving us obvious clues. Therefore the correct Strategy SS is "always switch". If we play this game 100 times adopting Strategy SS, we win roughly 99 times -- to be verified by simulation.

What about a strategy F -- "flip a coin to decide whether to switch"? I  feel this is less wise. The Red has very low potential, whereas the blue is an obvious suspect beyond reasonable doubt.

-----http://www.shodor.org/interactivate/activities/AdvancedMontyHall/ shows a parametrized version, with 10 doors. After you pick a door, 8 worthless doors open. My question is, if I follow "always-switch" i.e. Strategy SS, what's my chance of winning?

Answer: if my initial pick was right (unlikely -- 10% chance), then Strategy SS loses; if initial pick was wrong (likely -- 90%) then Strategy SS wins. Answer to the question is 0.90.

How about Baye's thereom?

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