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Two coin flips

>June 15, 2001
>
>painfully easy  aha:!
>
>i flip a penny and a dime and hide the result from you. "one of the coins came up heads", i announce. what is the chance that the other coin also came up heads?
>
>thanks tubby
>
>solution: easy
>
>
>Solved by David Grenier on June 17, 2001
>
>solution: painfully easy
>
>
>Assuming complete honesty on the part of the flipper, wouldn't the solution be 33%?
>
>There are four possible scenarios:
>
>HH
>TH
>HT
>TT
>
>Obviously the TT possibility can be discounted because it does not result in one of the two being heads.
>
>This lees us with three possibilities, only one of which has the other coin also being heads.
>
>Therefore one third.
>
>I think.
>
>I usually get these wrong.


You called it, you are wrong.

This is the sort of question I would dread getting asked
in an interview, because stupid people answer 50%,
smart people answer 33%, and *really* smart people
KNOW the answer is 50%.  ;-)

Let me propose a similar question, the answer to which
*is* 33%:

    i flip a penny and a dime and hide the result from
    you.  "Is at least one of the coins heads?"  you ask.
  "Yes" I truthfully reply.  what is the chance that the
    other coin also came up heads?

Do you see the subtle difference?  You have to allow
for the possibility of one of the four cases to get
eliminated, if I had responded "No".  The question, as
it is originally stated, does not allow for that.

To put it another way, suppose we do a number of
trials of the problem as it is originally stated.  Since I
am volunteering the information about the state of
one of the coins, I decide, UNBEKNOWNST TO YOU, to
always tell you the state of the dime.  "One of the
coins came up heads" I say if the dime came up heads,
and "One of the coins came up tails" I say if it
came up tails.  Your reply essentially amounts to
guessing the state of the penny then.  Are you telling
me that you can tell me you can do that with a better
than 50% accuracy?

If this question ever comes up in an interview, I see
myself getting into a heated argument with the
interviewer and completely blowing my chances.
Of course, anyone here is welcome to analyze the
above at the leisure and send me a job offer if they
agree I'm right.  ;-)


--Mark

Mark Schnitzius
Thursday, July 25, 2002

Even if we assume that the person who offers the information is completely neutral (an assumption that this groups is quite unwilling to make of the many fine people whose guidance counsellors suggested a career in piracy), there are at least two perfectly "reasonable" but contradictory possibilities:

1. In every case where there is at least one Heads, the person reveals this fact. (result: p=1/3)

2. The person always reveals that there is at least one Heads or at least one Tails (choosing between the two statements at random in cases where both are true). (result: p=1/2)

It isn't mere quibbling to say that there's something wrong with this problem.

And, the person may have some other rule for determining what information to give (or, indeed, whether to give any information at all).

If we want to turn this into a genuine probability problem, we need to remove the human element by forcing the person to obey some rule or other - and that rule must be given as part of the problem.

Steve Hutton
Thursday, July 25, 2002

Spot on analysis. 

>1. In every case where there is at least one Heads, the
>person reveals this fact. (result: p=1/3)
>
>2. The person always reveals that there is at least one
>Heads or at least one Tails (choosing between the two
>statements at random in cases where both are true).
>(result: p=1/2)

I would add to #1  "... and reflips if both coins are tails".

I maintain that in the absense of knowledge that the
flipper is eliminating cases, #2 is the better answer.  But
you're right, you can't say for sure, and the question is
poorly stated.

Mark Schnitzius
Thursday, July 25, 2002

I have to take issue with that bit about "really smart" people "knowing" that the answer is 50 percent. You can argue that the question is not well phrased but to say the answer is 50 percent based on your arbitrary choice of a strategy for the coin-watcher (for no strategy was indicated) is just wrong. I'd say that smart people realize that they are being asked an easy question on conditional probability which was translated into a word problem that can (with a bit of effort) be misconstrued.

The person who tells us that one coin was heads is not intended to be an actor in a game, he is just a plausible means of us learning the fact that one of the coins is heads. He is meant to be a "one of those coins was heads" flag. Is the question mathematically precise? No. Is the question clear to someone attempting to make a good faith interpretation? I think so. 

Alex Harris
Friday, July 26, 2002

In the absence of any other information apart from that stated in the original question, is it mathematically acceptable to say that the answer is "somewhere between 1/3 and 1/2 and we can only be more precise if we know the strategy of the coin-flipper"?

Adrian

Adrian Gilby
Monday, July 29, 2002

Depends on how much freedom you're giving the announcer. If you give him the option to say something else or even just to remain silent, then answer could be anywhere from 0 to 1. He could choose to say "one is heads" only when both are heads and just say "one is tails" or remain silent otherwise, or he could choose to say "one is heads" only when exactly one is heads and say "both are the same" or remain silent otherwise.

Alex Harris
Monday, July 29, 2002

Alex Harris wrote:
>I'd say that smart people realize that they are
>being asked an easy question on conditional
>probability which was translated into a word
>problem that can (with a bit of effort) be
>misconstrued.


Now that, I disagree with.  It's not an easy question --
many people misjudge it, or answer it correctly for the
wrong reasons, which is why it's an interesting problem.
Maybe it's become a chestnut of sorts, but I don't agree
that giving the chestnut response just because it's a
familiar problem is right.

I don't think I misconstrued the problem.  Assuming
that the flipper is eliminating possible outcomes for
you is a much bigger jump of reason than assuming
he's not.  I think the former would be less of a "good
faith interpretation" to someone who had never heard
the problem before.

All that said, the cocky nature of my original post was
intended to be humorous -- apologies if anyone was
offended.

Mark Schnitzius
Monday, July 29, 2002

Hye guys.
Well, I think either I m getting it wrong or all others have got it wrong. Considering the announcer is speaking the truth. There are basically 4 possibility.
HH  HT  TH  TT  .
Out of that..the, TT possibility is removed as the announcer says one of them is H.
Now there remains 3 cases.
HH  HT  TH. 
Consider 1st is H. then there is case,
          1) HH  & TH  for the other to have H. i.e. possibility is 2/3 out of HH HT TH.
          similirarly, consider 2nd is H ie. either HH or TH then also there is similar possibility 2/3 outof HH HT TH.

Write me back if you feel that I am getting it right & thats the correct ans.
C ya
Sachindshah@yahoo.com

Sachin d shah
Monday, September 02, 2002

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