Challenger: You Have Been Warned!
Dr. X is very upset that folks at the SDMB have been solving his probability puzzles. He sends you an E-Mail with the following challenge:
“Correctly solve my probability puzzle and I will give you both my 10 room house and my 10 room castle. Fail, and you become my slave. If you accept meet me at the house tomorrow.”
You’re not the type to back down from a challenge. But you are also no fool. Dr. X must be up to something! You send your spy to find Dr. X and get a copy of his plan.
Your spy somehow makes it back with a copy of Dr. X’s plan:
Dr. X will flip a fair coin. If heads, he will randomly place a clock in one room of his empty 10 room house. If tails he will randomly place a clock in one room of his empty, newly built (with help of Spoofe), 40 room mansion.
Then Dr. X will turn to you and let you select one room of the house for viewing, but not let you look in yet. Once he does this he will then show you two other empty rooms of the house. He will then give you these options:
- You may look in the room of the house you previously chose or
- You may look in another room of the house instead or
- You may look in a room of your choice in his 40 room mansion.
Then Dr. X will ask you to guess either the house or the mansion as the location of the clock.
Well, that’s the plan and you think it stinks. You know that unless you pick the option at the time that gives you the overall best odds of guessing correctly the location of the clock the fates will almost certainly cause you to lose.
Time to cheat! You call your spy and tell him to search the mansion after the coin flip. Then you’ll know without having to worry about no stinkin probabilities.
It’s the next day and you face Dr. X.
“I have fliped a fair coin,” says Dr. X. “If heads I have randomly put a clock in one room of the 10 room house. If tails I have randomly put a clock in one room of my 40 room mansion. You may choose a room of the house to look into now.”
You choose room #4.
“Fine.” says the sinister Doctor. “Now just to show you how fair I am let me show you the contents of rooms #8 and #9 of the house.”
They are of course empty. Duh.
“Now,” says the Doctor,“you may either look in room #4 as you wanted, or you may choose another room instead, or if you really want to I can show you one room of your choosing in my 40 room mansion.”
You stall for time. Where is your spy?? Suddenly your cell phone rings. “Excuse me, Dr. X,” you say. " I must take this call."
It’s your spy. Hah, his exotic bellydancer disguise must have gotten him into the mansion.
“I got into the mansion,” whispers your spy “but after I checked out Rooms #1, #2, #3, and #4 of the mansion, which were empty, my wig came off. I was recognized as a spy and tossed out. I hope I have helped.”
Incompetant spys again! Grrr!
You turn back to Dr. X.
“What’s it going to be?” sneers X. “Do you look in another room of the house, a room in the mansion, or stick with room #4? After you choose a room and look you will have to guess the location of the clock.”
Which do you choose in order to have the best probability of guessing correctly the location of the clock, and what is that best probability?