brain teaser

[SCfuji]

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]

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]

correctomundo. nice job.

[TomCollins]

What an interesting and original problem. I am truly enlightened. I'm glad such original posts occur on this board.