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Topic: Easy: Two Coin flips (Read 12637 times) |
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ootte
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Easy: Two Coin flips
« on: Jul 24th, 2002, 2:29am » |
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The chance is still 50%, since it doesn't matter what the other coin come up with.
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Oliver
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tealtoe
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I don't want to give away the answer right on this forum (unless you ask me to), but let me tell you that surprisingly your answer of 50% is wrong; this one is more difficult than it first appears.
Email me if you'd like the answer.
Ken.
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ootte
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Posts: 19
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Re: Easy: Two Coin flips
« Reply #2 on: Jul 25th, 2002, 6:43am » |
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on Jul 25th, 2002, 6:11am, tealtoe wrote:| but let me tell you that surprisingly your answer of 50% is wrong; this one is more difficult than it first appears. |
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If it is not some weird coin-throwing technique, and the coins are flipped one after another ... Umm. I don't know then.
Please post the solution or pm me, thanks.
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Oliver
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tealtoe
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I sent the solution to you via email.
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jmlyle
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As I see it, it is either a 50% chance (being an independent event), or the problem is about the veracity of the statment "One of the coins came up heads."
It seems that another "EASY" problem depends on a similar statement not being exclusive (i.e. "One" not meaning "Not two"). If the statement is a complete description of the entire state of things, then there is one answer. But if it is just a simple truthful declaration about a single coin (which seems entirely reasonable), then it should be 50%.... I think.
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tealtoe
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No, it certainly is not 50%, although you are on the right track--"One of the coins came up heads" is the key to the whole thing. If you don't get the answer, you can email me (kkitow "at sign" yahoo.com).
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Jonathan Wheaton
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As jmlyle noted, the statement "One of the coins came up heads" can be read as a description of the state of both coins or of only one. If the former, then it's more of a riddle; if the latter, then it's more of an exercise in probability. So which is it? I don't think there's a right answer to that one, although there's obviously only one right answer for each way of reading the statement. The riddle one is trivial, the probability one is harder (for me anyway).
In talking with one colleague, he looks at it this way. There are four states the two coins could be in. TT, HH, TH, HT. TT is out since you know one of them is heads. That leaves three possibilites, only one of which results in the second coin also being heads. Thus, 33% chance of the other coin being heads.
I'm not completely convinced, but it sounds reasonable. Thoughts?
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Mark Schnitzius
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Ha! Even if you consider "one of the coins came up heads"
to mean "AT LEAST one of the coins came up heads" (which
is how I believe it is meant), you are WRONG!
Here's how I responded to this question on another forum:
>June 15, 2001
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>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
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>solution: easy
>
>
>Solved by David Grenier on June 17, 2001
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>solution: painfully easy
>
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>Assuming complete honesty on the part of the flipper, >wouldn't the solution be 33%?
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>There are four possible scenarios:
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>HH
>TH
>HT
>TT
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>Obviously the TT possibility can be discounted because it >does not result in one of the two being heads.
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>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 one of my favorite problems, but I would dread
getting asked it in an interview, because stupid people
answer 50%, smart people answer 33%, and *really*
smart people KNOW the answer is 50%.  And I bet
they'd be expecting to hear 33%.
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 you knew that the flipper was eliminating all cases
of two tails being flipped, and only ever saying "At least
one of the coins is heads", then you could safely assume
the answer was 33%. But you don't know that the flipper
is doing that.
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 their leisure and send me a job offer if they
agree I'm right.
--Mark
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jmlyle
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Re: Easy: Two Coin flips
« Reply #8 on: Jul 26th, 2002, 8:19am » |
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So..... The problem that is presented is actually only able to happen 75% of the time after the double coin toss.....
And therefore, the coin state is NOT an independent event. It's possible state is dependent on the fact that the tosser was able to say what he said.
So, I think I know, but I guess I'll go with tradition and not post what I think is the answer. Could someone send me the answer, please (spam@myborg.com)?
--jmlyle
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Peter Seebach
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The point is, *IF* he says "at least one is heads", then you're in the 3 cases where you had at least one heads. It's not independant.
Comparison: If you do a dime and a penny, and say "the dime came up heads", the chances that the penny came up heads are now 50%. Neat, huh?
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Ion Rush
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Re: Easy: Two Coin flips
« Reply #10 on: Jul 27th, 2002, 3:00am » |
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tell me if this is going in the right direction
flip 2 coins, 4 possible outcomes
HH, HT, TH, TT
the fact that your buddy said one coin is heads means that TT must be removed, so you are down to 3 possible outcomes
HT TH HHs
now, if the question was, "what is the chance that BOTH coins are heads it would be 33%, however, the question is ,is the OTHER coin heads?
Now, as I see it, one coin is totally out of play and should no longer be influencing the outcome, only the OTHER coin does
(for example, your buddy flips the coins one after the other while shielding them from you. he only actually checked the first flip, and yes it was heads. he then flipped the second, and has not checked it yet. )
Because one of the coin's results is know is irrelevant, we are taking about one single coin, the OTHER coin, so it should be 50%
is this correct?
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Wah
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Re: Easy: Two Coin flips
« Reply #11 on: Jul 27th, 2002, 5:42am » |
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i don't think so
when he says one of the coins came up head, he could have meant either of the coins.
in the following 3 cases,his statement is true.
1st:T 2nd:H(the other coin is T)
1st:H 2nd:T(the other coin is T)
1st:H 2nd:H(the other coin is H)
therefor i think the chance are 1/3
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S. Owen
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Re: Easy: Two Coin flips
« Reply #12 on: Jul 27th, 2002, 7:09pm » |
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No, if you believe that the flipper is telling you that "at least one of the two coins came up heads", then the answer is indeed 1/3, for reasons outlined previously. This is probably the intended interpretation and the riddle should probably include the words "at least".
If the flipper is actually reporting on the state of only one coin, then that is a different scenario (to which the answer is 1/2, yes). But this of course is not the same as the "at least one" interpretation.
And in an interview, your best bet is to demonstrate your total understanding of the subtleties by asking for clarification, since *that* is actually the really smart answer.
on Jul 25th, 2002, 4:39pm, Mark Schnitzius wrote:
Ha! Even if you consider "one of the coins came up heads"
to mean "AT LEAST one of the coins came up heads" (which
is how I believe it is meant), you are WRONG!
...
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 you knew that the flipper was eliminating all cases
of two tails being flipped, and only ever saying "At least
one of the coins is heads", then you could safely assume
the answer was 33%. But you don't know that the flipper
is doing that.
--Mark |
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Mark Schnitzius
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Re: Easy: Two Coin flips
« Reply #13 on: Jul 30th, 2002, 8:25am » |
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on Jul 27th, 2002, 7:09pm, srowen wrote:| No, if you believe that the flipper is telling you that "at least one of the two coins came up heads", then the answer is indeed 1/3, for reasons outlined previously. This is probably the intended interpretation and the riddle should probably include the words "at least". |
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I disagree -- the answer is 1/2 in that case, UNLESS you make the assumption that the flipper keeps flipping until he gets at least one head, before telling you that at least one of the coins came up heads. That (to me) is a pretty big assumption.
on Jul 27th, 2002, 7:09pm, srowen wrote:| If the flipper is actually reporting on the state of only one coin, then that is a different scenario (to which the answer is 1/2, yes). But this of course is not the same as the "at least one" interpretation. |
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The flipper is ALWAYS reporting the state of only one coin. He needn't choose the dime or the penny every time; he might always choose which coin settled first, or which was closest to him, or whatever, but it's always the state of one coin. Even if he's always saying "at least one". Or maybe I'm misunderstanding you. Please explain to me the difference between reporting that "at least one" coin came up a certain way, versus reporting on the state of one coin (to which the flipper could always tack on "at least one" without lying).
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chopotter
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Posts: 2
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Re: Easy: Two Coin flips
« Reply #14 on: Jul 30th, 2002, 9:32am » |
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I've flipped both coins, and say "[at least] one of the coins is heads." If one of the coins is heads, there are only two possible states for the remaining coin, heads and tails, with heads having a 50% probability of being up.
There are not three possible outcomes, because the state of one of the coins has already been determined, i.e. you cannot use this coin to compute probabilites. There are only two possible outcomes. The remaining coin is heads or it is tails.
I'm quite ready to be disabused, but please do explain how this reasoning is faulty if you do.
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kenny
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Posts: 12
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Re: Easy: Two Coin flips (spoiler)
« Reply #15 on: Jul 30th, 2002, 11:34am » |
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In my opinion, you're all overlooking something. Here's the puzzle:
"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? "
It says nothing about the motivations of the flipper. Here are some possibilities:
1) Flip coins. If either or both comes up heads, say "one of the coins came up heads". Otherwise, say "one of the coins came up tails".
In this case, the probability of the other one being heads is 1/3.
2) Flip coins. Look at the dime. Say "one of the coins came up <facing on dime>".
In this case, the probability of the other one being heads is 1/2.
3) Flip coins. If either or both comes up tails, say "one of the coins came up tails". Otherwise, say "one of the coins came up heads."
In this case, the probability of the other one being heads is 1.
I think most arguments about the right answer hinge on assumptions about what the motivations of the tosser are.
-- Ken
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chopotter
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Re: Easy: Two Coin flips
« Reply #16 on: Jul 30th, 2002, 12:05pm » |
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If no assumptions are to be made, the riddle has no answer. Actually, I guess it has an answer for each set of assumptions. How... messy. I'd prefer there to be only _one_ answer. 
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josie
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I read it as only one coin came up heads. The question asks, "what is the chance that the other coin also came up heads?" which implies that the "other" coin is the one that didn't come up as heads. So, I think zero chance.
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jc
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Re: Easy: Two Coin flips
« Reply #18 on: Aug 3rd, 2002, 10:06pm » |
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Yeah, I also got 1/3. Since it asks what the chances are that the "other" coin was also heads, it implies, "what's the chance that both were heads?". Since there are 4 possibilites, and you know tails/tails isn't one of them since it needs at least one heads, heads/heads is one of three possible answers. So 1/3. 
It's much easier if you don't overcomplicate.
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zoobiewa
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Re: Easy: Two Coin flips
« Reply #19 on: Aug 4th, 2002, 11:02am » |
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I know the answer. First we should put the original question out there:
"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?"
Then we need the table of all possible outcomes
tt th ht hh
The possibility of tt happening is impossible, so we can x that out. If you were randomly flipping to see if a pattern emerged, you would not be able to use any outcomes that came up with tt in them. This weighs the results, so now we have only three available options, which are: th ht and hh. With th and ht, they are the same thing, which means there are really only two options, one heads and the other tails, or both heads. That means there is a 50/50 chance.
I tried this by flipping a penny and a dime 35 times, and tossing out any touble tails. After 35 throws, they were almost neck and neck, with ht winning by one point. It is 50%
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jc
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on Aug 4th, 2002, 11:02am, zoobiewa wrote:| With th and ht, they are the same thing, which means there are really only two options, one heads and the other tails, or both heads. That means there is a 50/50 chance. |
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Nope, sorry. They're not the same thing. He doesn't tell you which coin already landed heads. th would mean, say, that the penny is tails and the dime is heads, whereas ht would mean the penny is heads and the dime is tails. They're both viable possibilities.
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S. Owen
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Re: Easy: Two Coin flips
« Reply #21 on: Aug 5th, 2002, 6:19am » |
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on Jul 30th, 2002, 8:25am, Mark Schnitzius wrote:
I disagree -- the answer is 1/2 in that case, UNLESS you make the assumption that the flipper keeps flipping until he gets at least one head, before telling you that at least one of the coins came up heads. That (to me) is a pretty big assumption. |
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Sure, you are given in this riddle that it happens that at least one of the two coins came up heads. If you want to say that as "the flipper kept flipping until getting a head", OK.
on Jul 30th, 2002, 8:25am, Mark Schnitzius wrote:
The flipper is ALWAYS reporting the state of only one coin. He needn't choose the dime or the penny every time; he might always choose which coin settled first, or which was closest to him, or whatever, but it's always the state of one coin. Even if he's always saying "at least one". Or maybe I'm misunderstanding you. Please explain to me the difference between reporting that "at least one" coin came up a certain way, versus reporting on the state of one coin (to which the flipper could always tack on "at least one" without lying). |
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I am saying the difference between the two interpretations is this:
A) "At least one of the coins is heads."
Possible outcomes were: HT TH HH
B) "I only looked at one coin (chosen according to my whim) and it is heads."
Possible outcomes were: HT HH
The difference in what the interpretation tells you about the possible outcomes of the flips makes the difference.
Or to put it in the terms you used, it is the difference between flipping two coins until at least one is heads, and flipping two coins, selecting one to look at, and continuing until that selected one is heads.
You're right that it doesn't matter how the flipper selects which coin to report on, under interpretation B. But it does matter if she is only giving you information about one coin, because it affects your information about the outcomes.
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| « Last Edit: Aug 5th, 2002, 6:21am by S. Owen » |
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anshil
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Posts: 41
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Re: Easy: Two Coin flips
« Reply #22 on: Aug 5th, 2002, 7:40am » |
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The answer is 50%. No I did not think about it, I just wrote a little pyhton program.
from random import *
t = 0; # number of cases the condition was true
f = 0; # number of cases the condition was false
while 1:
a = choice(range(2)) # flip the first coin
b = choice(range(2)) # flip the second coin
if a == 0 and b == 0: # if both are tail ignore and reflip
continue;
if a == 1 and b == 1: # if both are head the condition is true
t = t + 1
else: # otherwise the other coin is not head
f = f + 1
if f > 0: # just don't device by zero
print "t=", t, " f=", f, " ratio=", 1.0 * t / f
#end of program
It will print out 0.5 very fast, every time I run. So the answer is 50% as long I've not coded something wrong. Now theorize so long until you get the answer the expirment showed. (science, believe the expermient, not the theory)
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anshil
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Re: Easy: Two Coin flips
« Reply #23 on: Aug 5th, 2002, 7:45am » |
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Damm am I stupid, of course the answer is not 50%.
2/3 and 1/3 are actually true, if I devide false by truth I get 0.5, so truth is twice as often as false. so 33% is correct!
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peter johnson
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>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?
The chance of the second coming up heads is zero, 0%. ONE of the coins came up heads !
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