The evil Dr. X is annoyed that his previous puzzle was solved, so he offers a new wager. He’ll flip another fair coin. If it is heads, he’ll put a clock in a random room of his 10 room house. If it comes up tails he’ll put the clock in a random room of his 10 room castle.
Dr. X tells you he’ll be sporting. After he flips and places the clock he will actually show you the contents of 5 of the 10 rooms in the house. That’s five rooms of HIS choosing, by the way. Then you’ll make your wager as to the location of the clock–“castle” or “house”.
You agree. You, of course, do not trust Dr. X. While the coin and rules are legit, he will most certainly show you 5 empty rooms in the house.
You agreed because you have an ace up your sleeve. Your spy has infiltrated the castle of Dr. X dressed in an inconspicuous suit of armor. Your spy will check the rooms and report to you.
The contest begins. Dr. X shows you rooms 2, 5, 7, 8, and 10 in the house. They are empty.
Just then the phone rings. Cursing, Dr. X goes to answer it, giving you time to peek into room 6. It is empty.
The call is for you. It is your spy. “I checked 3 rooms of the castle, but then I was caught and thrown out,” he whispers over the phone. “All 3 rooms I saw were empty.”
Dr. X now asks you to wager. Is the clock in the house or in the castle?
A. Which do you pick?
B. What is the probability you are right?
ok, there are 20 rooms in total. you have seen 9 of them (6 in the house and 3 in the castle) leaving 11. there are 4 unseen rooms in the house. there are 7 unseen rooms in the castle. The coin flick is irrelevant, as there are even odds for it appearing in either A) the house and B) the castle.
there is a 50% chance 4 out of the 11 remaining rooms, and a 50% it is in one of the 7 remaining rooms.
You can pick either. you have as much chance of being right.
so, I would pich the Castle, simply there are more rooms that he could have put it in, but the odds are still 50% for either option
Don’t do it! Dr. X, no fool he, realizes that some folks might try such a theiving stunt. That’s why his wallet is booby-trapped. When the mind-warping rays from the lethal leatherware finish with you, you are nothing but a shell of your former self–fit only to mindlessly slave away on Dr. X’s new living quarters–a dank, dark, 40-roomed mansion.
Dr. X is, however, miffed because one poster did figure out the puzzle correctly. That poster is , of course,… is…Arrgh…Damn…Stop…ARrghh…Curse you Dr. X!..
The probability is 8/15 that the clock is in the house.
[ul][li]At the beginning, there are twenty rooms, each of which have a 1/20 chance of containing the clock. There is a 50/50 chance the clock is in either the house or the castle.[/li][li]You know that the five rooms Dr X shows you will be empty, wherever the clock is. So you have no further information on whether the clock is in the house or castle, but you do know that five particular rooms are empty. Thus, the five remaining rooms of the house each have a 1/10 chance of containing the clock, while the ten castle rooms have a 1/20 chance apiece.[/li][li]You peek in room 6, and it’s eliminated. Thus, the four remaining rooms of the house each have a 1/9 chance of containing the clock, while the ten castle rooms have a 1/18 chance apiece.[/li][li]The spy reports on three rooms of the castle, eliminating them. Thus, the four remaining rooms of the house each have a 2/15 chance of containing the clock, while the seven remaining castle rooms have a 1/15 chance apiece.[/li][li]Add 'em up: there’s a 8/15 chance the clock is in the house.[/li][/ul]
This process is easier to to through with a pencil and paper, eliminating the possibilities in sequence. You just have to realize that the first bit of information (Dr X’s five rooms) changes the relative probabilities of the clock’s location, while the second two bits do not.
