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SCfuji
11-08-2004, 05:57 AM
Your friend asks you to guess a number between 1 and 3. You tentatively guess 1, and your friend tells you the number is not 2 (intentionally choosing one of the numbers you did not guess). The chances that the number is 3 are then 2/3.

Friend A asks you to guess a number between 1 and 3. You and Friend B work out a plan for you to initially choose 1, while Friend B guesses either 2 or 3 in order to take advantage of the above probability. Friend B guesses 2, and Friend A tells Friend B that he is wrong. Now the chances that the number is 3 are 1/2.
Explain.

young nut
11-08-2004, 08:46 AM
This is just another word problem stemming from that gameshow with Monty Hall, where you had to guess between 3 doors.

In the first example, your probability of your first choice being correct is 1/3. Now when he says that one of the other numbers is not it, you should always switch because him revealing this information gives a 2/3 chance of the other number being correct.

with the second example, your odds are effectively changed because the second person.

numbers 1 2 3 are choices each with a probability of being 1/3 correct. If you both pick numbers that are incorrect, your odds of your choices then go to 1/2. The reason they do not change to 2/3 as the previous example is because of the extra information you have about that second number.

Take for example the 3 doors:
Door A, B, and C.

I choose A.
bobby chooses B.

We are told that there is nothing behind door C. The host's only option here was to either 1) reveal a prize if the right door was chosen or 2) tell you that the unchoses door did not have a prize. He lacked the extra option of 3) telling you that one of the remaining doors (EITHER B or C) was not correct. This extra option is what transfers the other 1/3 probability to that other door. But when he only has the option to reveal information about one unchosen door, the odds get split evenly between the remaining two doors.

SCfuji
11-08-2004, 07:00 PM
correctomundo. nice job.

TomCollins
11-09-2004, 11:03 AM
What an interesting and original problem. I am truly enlightened. I'm glad such original posts occur on this board.