

Probability Riddle--are you a "halfer" or a "thirder"? - Raphael
http://www.maproom.co.uk/sb.html

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alex_c
The way I read it, the problem is ambiguous.

1/2 is the answer to "What is the probability that the coin landed Heads?"

1/3 is the answer to "What is the average probability of you getting it right
if you say the coin landed Heads?"

So I guess that makes me a "Can you rephrase the question?-er"

~~~
psyklic
Definitely.

If your goal is to maximize the number of correct guesses _across all
questionings_ , then always answer 1/3.

If your goal is to guess the _current coin flip_ as accurately as possible,
then always answer 1/2. (I interpret the question as this.)

~~~
emmett
Try thinking about it this way instead:

If the coin comes up tails, you will be asked 1,000,000 times in a row. If the
coin comes up heads, you have a 1/1,000,000 chance of being asked.

I wake you up. If you had to guess if the coin was heads or tails, what would
you say?

~~~
trevelyan
There is a 50% chance of the coin landing on Heads, and a 50% of it landing on
Tails.

The events in question are not independent. You have a 50% chance of being
woken once, and a 50% chance of being woken 1,000,000 times.

Therefore the correct answer is 50%.

~~~
GavinB
I guess the question is whether being wrong 1,000,000 times is equivalent to
being wrong 1 time.

Sadly, the problem doesn't define that.

------
GavinB
On Heads, sleeping beauty gets asked once. On Tails, she gets asked twice. If
I walk into the room and see her being asked, it's a better bet that the coin
landed tails.

If this experiment is repeated many times, "Tails" will be the right answer
twice as often.

You can try this with a friend. As someone pointed out later on in the page,
you just answer the first question freely and if it's tails are required to
answer the second question the same way as you did the first. Or just answer
heads every time--no memory wiping required. If it's really 50/50 then
answering heads should get you your answer.

Because the interviewer asks questions in an unbalanced manner, the
interviewee actually gets more information just by being asked.

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cousin_it
Wanna fry your brains? Some time ago I was intrigued to learn about the non-
probabilistic variant of the two envelopes problem:

[http://en.wikipedia.org/wiki/Two_envelopes_problem#A_non-
pro...](http://en.wikipedia.org/wiki/Two_envelopes_problem#A_non-
probabilistic_variant)

------
hugh
In a slightly more evil thought experiment (Harvey Dent might approve), we
could make one slight change: if it's heads we interview her once and then let
her go, but if it's tails we'll keep on waking her and interviewing her every
day for a thousand years.

I think this version makes it clearer: if it's heads she'll be interviewed
once, if it's tails she'll be interviewed thousands of times. At any given
interview she should assign a very small probability to the possibility of
heads.

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Raphael
I am a halfer.

    
    
    	 1/2      1/2
    	 /  \ 	   |
    	Mon Tue   Mon
    	1/2 1/2    1
    	 T   T     H
    
    

I think I have identified the thirder problem. They are assuming that it's
Monday:

    
    
              Mon     Tue
              3/4	  1/4
              /  \	   |
             2/3 1/3   T
             |    |
             H    T

~~~
GavinB
Your first diagram makes it look like something is split between Monday and
Tuesday, when it's not split at all. You get both of those "tails" interviews,
in sequence. So you get asked two times for the answer when it is tails, and
only once when it is heads.

Try doing the experiment several times and see how well you do picking heads
every time you "wake up." There's no need to have memory alteration if you set
your strategy beforehand. You'll only be right 1/3 of the time if you bet on
heads.

~~~
Raphael
There is a split. Given tails, it is either Monday or Tuesday.

~~~
GavinB
I think you may be misreading the question. Given tails, you are asked both
Monday AND Tuesday. That's why it gets tricky--you have to double down if it's
tails.

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sysop073
I feel like I must be missing something, although I suppose all good paradoxes
give you that feeling. If it's heads, she's asked once (Monday) what it was,
if tails, she's asked twice (Monday and Tuesday). Therefore, if she has no
idea which of the three states she's currently in, there's a 1 in 3 chance it
was heads

~~~
psyklic
There are three states, but they are not equiprobable. One-half of the time
she'll be asked once, and one-half of the time she'll be asked twice. Hence,
the one-half argument is because she'll only be in the "once" (Heads) state
1/2 of the time.

~~~
zasz
They are equiprobable. When the coin is flipped, it is 50% likely that it will
be heads or tails. When it's tails, Beauty must be interviewed again.
Therefore, the number of times she's in the state Monday+tails is equal to the
number of times she's in the state Tuesday+tails. However, the number of times
she's in the state Monday+tails is equal to the number of times she's in the
state Monday+heads. No other possible states can occur, so these three states
are equiprobable.

The answer is that at any point in time, absent other information, the coin is
equally likely to be heads or tails, but as soon as Beauty receives the new
information that she's being interviewed, it changes the known information,
making it 1/3 instead of 1/2.

~~~
psyklic
The three states are not equiprobable. Suppose that she must say instead which
particular state she is most likely to be in: M+T, T+T, or M+H. Well, we know
that Heads is chosen 1/2 of the time, so she should choose M+H! After all, the
probability she is in M+T is only 1/4, and the probability she is in T+T is
only 1/4. (Since Tails is chosen 1/2 of the time.)

------
Eliezer
I don't know.

