Cross Post (Off Topic) Midnight 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.

[Chris Daddy Cool]

monty hall

[SCfuji]

wrong answer. /images/graemlins/tongue.gif

[MrFeelNothin]

Alright, so its 430 in the morning and Ive accidentally deleted this post twice while trying to get my thoughts down correctly. Here goes again:
In the first situation, initially Hero has a 1/3 chance of guessing correctly initially and 2/3 of guessing incorrectly. Obviously. However whether Hero was right or wrong Villian would choose one of the other numbers as you said here: [ QUOTE ]
intentionally choosing one of the numbers you did not guess

[/ QUOTE ] SO 1/3 of the time Hero was correct initially and Villian said 2 was incorrect to throw Hero off. 2/3 of the time Hero was incorrect and therefore the only remaining option, 3, is correct. The probabilities have not changed since Hero guessed, there is still a 1/3 chance Hero was correct, and a 2/3 chance Hero was incorrect and that therefore 3 is correct. Do you see why? /images/graemlins/tongue.gif

In the second situation Hero and his cohort are not "taking advantage of the above probability." The only reason the above probability existed is because of this key sentence which I will post again:
[ QUOTE ]
intentionally choosing one of the numbers you did not guess

[/ QUOTE ]
In this second situation Villian has simply told Heros friend he is wrong, leaving a 1/2 probability for each of the remaining two options. UNLESS Villian was doing the opposite of what he did in situation one, where whenever Hero or his friend were correct, he would tell the other they were wrong and if they were both wrong he would randomly tell one. In that case I believe the probability of 3 would be 1/3, for the same reason as situation 1: there was a 2/3 chance initially that one of the two heroes was correct and a 1/3 chance they were both wrong and the probabilities have not changed as long as he followed that system. Half the time Heros friend would be the one who would be told he was wrong, but 2/3 of the time this would be b/c the original Hero was correct, because Heros initial probability of being correct is 1/3. 33.3/50=2/3. Or a 1/3 chance that 3 is correct.

How did I do Fuji? I know it made sense in my head but man that was tough to wrap my brain around. Dont know if it actually makes sense in my post or not. /images/graemlins/crazy.gif

[waffle]

The difference is in who is choosing the number the host tells you extra information about. In the first example, the host always chooses a number that is not the chosen number. In the second example, your friend is just picking a number to ask about.

In the second example, all you know is that the second number is not it, so there's a 50/50 chance of it being 1 or 3.


Regarding the case where the hosts tells you about number 2.

If you pick 1 and the chosen number is 3, the host will have to tell you about number 2 every time.

If you pick 1 and the chosen number is 1, the host will only tell you about number 2 half the time.

I suspect this is where the difference comes in.