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#1
Old 03-20-2016, 05:54 PM
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Frog riddle: does the answer make sense?

Based on this video.

Here's the situation. You're in a jungle. You've accidentally eaten a poisonous mushroom. But you know how to get the antidote. There's a species of frog that lives in the jungle and if you lick one it's an antidote for the poison. And the frogs have no problem being caught or licked.

The problem is only female frogs produce the antidote. Licking a male frog does you no harm but it doesn't help you either. And male and female frogs look identical. The only readily discernible difference between them is male frogs make a croaking sound which females do not. Males and females occur in equal numbers and there's no pattern to their grouping or their croaking.

You're lucky. You see a stump twenty feet ahead of you with a lone frog sitting on top of it. But then you hear a frog croaking behind you. You turn and see two frogs sitting on a rock twenty feet behind you. You can tell it was one of those two frogs which croaked but you can't tell which one. There are no other frogs in the area.

You start to get dizzy and you realize you only have enough time to walk to either the stump or the rock before you pass out and die. You can go to the stump and lick that frog or go to the rock and lick both frogs. So which one do you go to?

My answer:
SPOILER:
It doesn't matter. You have a 50/50 chance either way. There's a 50% chance the frog on the stump is female. There are two frogs on the rock but you know one of them is a male even if you don't know which one it is. So essentially you have a single frog which might be female and that gives you the same 50% chance.


Here's the official answer:
SPOILER:
You should go to the rock because you have better odds. There are four possible groupings of two frogs: FF, FM, MF, MM. You heard the croak so you know this pair isn't FF. But of the remaining three groupings, two contain at least one female. So your odds are 2/3 in your favor.
#2
Old 03-20-2016, 06:16 PM
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I think that your answer is correct. We're told that there's no pattern to the croaking, but presumably, one is still twice as likely to hear a croak from a pair of males than from a single male and a female. So the fact that you heard a croak not only eliminates the FF possibility, but it also weights the MM possibility higher.
#3
Old 03-20-2016, 06:33 PM
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Quote:
Originally Posted by Chronos View Post
We're told that there's no pattern to the croaking, but presumably, one is still twice as likely to hear a croak from a pair of males than from a single male and a female.
Unless males only croak in the presence of females.
#4
Old 03-20-2016, 06:34 PM
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It's really just a restatement of the Boy/Girl Paradox.

https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
#5
Old 03-20-2016, 07:51 PM
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If you see a pair of frogs on the rock because you heard one and only one croak, there are four possible pairings.

Female on the left and a Croaking Male on the Right - you live
Croaking Male on the Left and Female on the Right - you live
Croaking Male on the Left and Silent Male on the Right - you die
Silent Male on the Left and Croaking Male on the Right - you die

So it is still 50/50.
#6
Old 03-20-2016, 07:53 PM
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Except that the boy/girl puzzle is extremely sensitive to precise wording, and so it's very easy to make a variant of it for which the answer is different. And the reason why the answer is so counterintuitive is that the precise situation which leads to the "paradoxical" answer is extremely rare in real life. In fact, it's quite difficult to contrive a situation in which the paradoxical answer actually is correct.
#7
Old 03-20-2016, 08:27 PM
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Yeah, it's 50/50, and newme explains why in a concrete way.

From a more abstract point, it reminds me of the whole Monty Hall thing. The trick to that one is to remember that Monty has to show you a goat. He has no choice in the matter. Removing choice from a probability situation fundamentally changes the default probabilities.

The frog croaked. You had no effect on that. Without the croak your options are clearly better two lick two frogs rather than one. But now you know that at least one of those frogs is a male. Which means the frog can just be simply removed from consideration. Therefore, you have two frogs, one in each direction, and each has an equal chance of being male or female as the problem is laid out.

The people who create these scenarios have a weird tendency to out-clever themselves.

Last edited by Hoopy Frood; 03-20-2016 at 08:28 PM.
#8
Old 03-20-2016, 09:38 PM
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Quote:
Originally Posted by newme View Post
Female on the left and a Croaking Male on the Right - you live
Croaking Male on the Left and Female on the Right - you live
Croaking Male on the Left and Silent Male on the Right - you die
Silent Male on the Left and Croaking Male on the Right - you die
Are you sure those four possibilities are equally likely? My intuition would be that the two male/male cases are half as likely as the other cases.

Anyway the real answer is that you should go to the two frogs, because either you're right and it's 50/50, or you're wrong and it's better than that.
#9
Old 03-20-2016, 09:42 PM
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Quote:
Originally Posted by Hoopy Frood View Post
The frog croaked. You had no effect on that. Without the croak your options are clearly better two lick two frogs rather than one. But now you know that at least one of those frogs is a male. Which means the frog can just be simply removed from consideration. Therefore, you have two frogs, one in each direction, and each has an equal chance of being male or female as the problem is laid out.
That's pretty much what I figured.

Another way to look at it is to suppose that you heard a parrot squawk instead of hearing a frog croak and you turned around in response to that. And when you turned around there was the parrot on a rock with a single frog next to it. In this scenario, you'd have one silent frog on the stump and one silent frog on the rock. Your choices would clearly be equal.

And this scenario is essentially the same as the original scenario. If female frogs are silent, then turning around to look at a croaking frog is no different than turning around to look at a squawking parrot. In neither scenario did you have any reason to expect to see a silent frog when you turned around.
#10
Old 03-20-2016, 09:52 PM
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That's not the same scenario, because you can't identify which frog croaked in the real scenario.

1/3 of the time, there's a female on the left.
1/3 of the time, there's a female on the right.
1/6 of the time, there's a silent male on the left.
1/6 of the time, there's a silent male on the right.
#11
Old 03-20-2016, 10:15 PM
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The "official answer" is wrong because it says there are 4 possible groups, ff,mf,fm,mm.
The problem there is they differentiate male on left, female on right and vice versa for the MF/FM possibilities, but not for the mm or ff ones. If they're going to count both mf and fm as seperate possibilites, then they need to include m1m2 and m2m1, and f1f2 and f2f1 as well. Doing that, and then elimitating both ff groups leaves 2 possible mm groups and 2 possibile mf groups, so 50/50.
#12
Old 03-20-2016, 10:23 PM
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But the mm possibility really is half as likely as one male and one female. If you flip two coins, you'll get two heads half as often as you get a head and a tail.
#13
Old 03-20-2016, 11:14 PM
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The lone frog on the stump did not croak. It was either a female frog or a male frog. But the chance is better that it is a female frog because if it was a male frog, it might have croaked, while a female frog could not have croaked, right?
#14
Old 03-20-2016, 11:18 PM
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And that should apply to the two frogs on the rock as well. A silent frog is more likely to be female because some male frogs choose not to be silent, while all female frogs must be silent.
#15
Old 03-20-2016, 11:34 PM
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The official answer is correct.

You might have a silent male on the left, or a silent male on the right, but you also know that you do NOT have either two silent males or two croaking males on the rock.

Last edited by Biotop; 03-20-2016 at 11:34 PM.
#16
Old 03-20-2016, 11:36 PM
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Originally Posted by borschevsky View Post
Anyway the real answer is that you should go to the two frogs, because either you're right and it's 50/50, or you're wrong and it's better than that.
I like this answer.

Also, I know it's fighting the hypothetical, but I don't see, practically, how you could have known the croak had to have come from one of the two frogs on the rock; how could you possibly know there wasn't a third frog behind (or otherwise nearby) the rock?
#17
Old 03-20-2016, 11:54 PM
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Originally Posted by Thudlow Boink View Post

Also, I know it's fighting the hypothetical, but I don't see, practically, how you could have known the croak had to have come from one of the two frogs on the rock; how could you possibly know there wasn't a third frog behind (or otherwise nearby) the rock?
Does a frog's croak echo?
#18
Old 03-21-2016, 12:05 AM
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Originally Posted by borschevsky View Post
That's not the same scenario, because you can't identify which frog croaked in the real scenario.

1/3 of the time, there's a female on the left.
1/3 of the time, there's a female on the right.
1/6 of the time, there's a silent male on the left.
1/6 of the time, there's a silent male on the right.
This. Well explained.
#19
Old 03-21-2016, 01:06 AM
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Quote:
Originally Posted by borschevsky View Post
That's not the same scenario, because you can't identify which frog croaked in the real scenario.

1/3 of the time, there's a female on the left.
1/3 of the time, there's a female on the right.
1/6 of the time, there's a silent male on the left.
1/6 of the time, there's a silent male on the right.
No, I think you're putting up the wrong odds. There's no reason to assume a female is twice as likely as a silent male, which is what you're doing.

You know that one frog made a noise (and is therefore male) and one frog did not.

So you start with this:

50% chance left frog croaked (male) and right frog was silent (male or female)
50% chance right frog croaked (male) and left frog was silent (male or female)

You also know that there's an equal chance that a silent frog is male or female.

So you refine your odds:

25% chance left frog croaked (male) and right frog was silent and male
25% chance left frog croaked (male) and right frog was silent and female

25% chance right frog croaked (male) and left frog was silent and male
25% chance right frog croaked (male) and left frog was silent and female

Now you divide up the possibilities with and without a female:

25% chance left frog croaked (male) and right frog was silent and male
25% chance right frog croaked (male) and left frog was silent and male

25% chance left frog croaked (male) and right frog was silent and female
25% chance right frog croaked (male) and left frog was silent and female

You have a 50% chance that one of the frogs on the rock is female.

Last edited by Little Nemo; 03-21-2016 at 01:06 AM.
#20
Old 03-21-2016, 01:15 AM
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Originally Posted by Little Nemo View Post
Another way to look at it is to suppose that you heard a parrot squawk instead of hearing a frog croak and you turned around in response to that. And when you turned around there was the parrot on a rock with a single frog next to it. In this scenario, you'd have one silent frog on the stump and one silent frog on the rock. Your choices would clearly be equal.

And this scenario is essentially the same as the original scenario. If female frogs are silent, then turning around to look at a croaking frog is no different than turning around to look at a squawking parrot. In neither scenario did you have any reason to expect to see a silent frog when you turned around.
Consider this scenario presented with different wording.

You heard a noise behind you. You turned around to look at the object which made the noise. You knew before turning around that the object, whatever it was, was not a female frog because female frogs are silent.

When you turned and saw the object which made the noise, you saw a silent frog sitting on the rock next to the object. You know there is a 50/50 chance this frog is female.
#21
Old 03-21-2016, 01:23 AM
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Why would you assume a silent frog is equally likely to be male as female? If a male sometimes croaks and the female never croaks, and male frogs and female frogs are evenly distributed, then a non-croaking frog is more likely to be female.
#22
Old 03-21-2016, 02:09 AM
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Possibilities of a random 2 two frogs --- assuming a male is just as likely to croak as not:

1. Male (L), Female (R) --25%
Which is either:
Silent Male (L), Female(R) --12.5%
or
Croaking Male (L), Female(R) -- 12.5%

2. Female (L), Male (R) --25%
Which is either:
Female (L), Silent Male (R) --12.5%
or
Female (L), Croaking Male (R) --12.5%

3. Female (L), Female (R) --25%

4. Male (L), Male (R) --25%
Which is one of these four:
Silent Male (L), Silent Male (R) --6.25%
Croaking Male (L), Croaking Male (R) --6.25%
Silent Male (L), Croaking Male (R) --6.25%
Croaking Male (L), Silent Male (R) -- 6.25%

****
Now you hear a single croaking frog and turn to see two frogs. The chance is two out of three that one of these frogs is female.

****

However, it should be noted on this basis that the single non- croaking frog on the rock is therefore 75% likely to be female. So you should really lick that one instead!
#23
Old 03-21-2016, 02:20 AM
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Er... scratch that. The single frog is also 2 out of 3.

So now it seems to me it doesn't matter which you do. The odds are 67% in your favor either way you lick.
#24
Old 03-21-2016, 03:05 AM
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After thinking more about this, I have a new answer depending on the following conditions.

1.) A male frog will croak at most once in the time allowed.

2.) All male frogs paired with another frog have an equal chance of croaking, and that chance is not affected by the sex of the other frog.

3.) You cannot tell if the croaking you hear from the pair on the rock is from one or both frogs.

4.) I am assuming a frog by itself will not croak cuz it hurts my brain to think about it otherwise.

If the odds of a male frog croaking in the allowed time interval are 1/n, where n is greater than or equal to 1, then if you look at all possible combinations, you will survive 2n times, die 2n-1 times, so the odds of surviving by licking the pair are

2n/(4n-1).

------------------------------------------------------------------------------------------------
So, in the case where a paired male always croaks, then n=1, and the odds of survival are 2/3 :

FemaleFemale (Discarded because you never hear anything)
FemaleMale - Survive
MaleFemale - Survive
MaleMale - Die

---------------------------------------------------------------------------------------------------
In the case where the odds of a paired male croaking are 1/2, then the odds of surviving are 4/7:

M=croaking male
m=silent male
F=female that would have croaked had it been male
f=female that would have been silent had it been male

ff - discarded
fF - discarded
Ff - discarded
FF - discarded

fm - discarded
fM - survive
Fm - discarded
FM - survive

mf - discarded
mF - discarded
Mf - survive
MF - survive

mm - discarded
mM - die
Mm - die
MM - die


Survive 4 times, die 3 times, odds of surviving are 4/7.

---------------------------------------------------------------------------------------------------

In the case of the odds of a paired male croaking are 1/10, in your head you can work out that, in 400 possible combinations

you will survive 20 times

you will die 19 times

you will discard the remaining 361 cases where you hear nothing

Odds of surviving drop to 20/39; you are starting to approach 50:50.

--------------------------------------------------------------------------------------

If the odds are extremely high against a paired frog croaking, then
as n approaches infinity, limit{2n/(4n-1)} = 1/2


So, worst case, it is 50:50 if you lick the pair, but could be 2 outta 3.
#25
Old 03-21-2016, 03:15 AM
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Quote:
Originally Posted by Biotop View Post
Why would you assume a silent frog is equally likely to be male as female? If a male sometimes croaks and the female never croaks, and male frogs and female frogs are evenly distributed, then a non-croaking frog is more likely to be female.
Because those are the basic premises of the puzzle.
#26
Old 03-21-2016, 07:44 AM
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Originally Posted by Little Nemo View Post
Because those are the basic premises of the puzzle.
The video puzzle is obviously trying, as wolfman said, to simply restate/redo the classic boy/girl puzzle. However, the problem is not so simple as given because we don't know how often male frogs croak.

If you know, because I tell you, that at least one of two frogs is male, than yes the odds are two of three that the other is female. But here you know because of something a male frog might do. That changes things.
#27
Old 03-21-2016, 08:26 AM
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Originally Posted by Biotop View Post
Why would you assume a silent frog is equally likely to be male as female? If a male sometimes croaks and the female never croaks, and male frogs and female frogs are evenly distributed, then a non-croaking frog is more likely to be female.
I'm already lost in all the logic trees, but I'd just like to point out that this is irrelevant for the purposes of the puzzle, because a non-croaking frog exists in both directions.
#28
Old 03-21-2016, 08:42 AM
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Originally Posted by Dead Cat View Post
I'm already lost in all the logic trees, but I'd just like to point out that this is irrelevant for the purposes of the puzzle, because a non-croaking frog exists in both directions.
Right, but a male frog could have croaked . Some of the male frogs you meet will croak. Some will not. None of the female frogs will croak. You will over time meet the same number of male frogs as female. Therefore, any particular non-croaking frog you meet has a higher chance of being female.
#29
Old 03-21-2016, 09:00 AM
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Clearly, without any knowledge of the gender of the frogs, choosing the two frogs wins 75% of the time.

Leaving aside the issue of croaking for the time being, if we somehow magically know that at least one of the pair of frogs is male (like in the Boy/Girl Paradox) then this chance falls to ~67%, as in the official answer.

The frequency of croaking is a problem. For someone to have noted that only male frogs croak, they must croak sometimes. But the OP says there is no pattern to the croaking. I think the only sensible way to interpret this is to assume that non-croaking does not imply anything about the gender of a frog. Not strictly true, as Biotop says, but if male frogs croaked often, that would constitute a pattern. So perhaps we should say that they croak so infrequently, once every ten years perhaps, that it does not materially affect the probabilities here. That is, the croak is merely this puzzle's way of letting you know that the pair of frogs are not both female. Under that interpretation, the official answer stands.
#30
Old 03-21-2016, 09:17 AM
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Originally Posted by Jason D View Post
Clearly, without any knowledge of the gender of the frogs, choosing the two frogs wins 75% of the time.

Leaving aside the issue of croaking for the time being, if we somehow magically know that at least one of the pair of frogs is male (like in the Boy/Girl Paradox) then this chance falls to ~67%, as in the official answer.

The frequency of croaking is a problem. For someone to have noted that only male frogs croak, they must croak sometimes. But the OP says there is no pattern to the croaking. I think the only sensible way to interpret this is to assume that non-croaking does not imply anything about the gender of a frog. Not strictly true, as Biotop says, but if male frogs croaked often, that would constitute a pattern. So perhaps we should say that they croak so infrequently, once every ten years perhaps, that it does not materially affect the probabilities here. That is, the croak is merely this puzzle's way of letting you know that the pair of frogs are not both female. Under that interpretation, the official answer stands.
This is where I am still struggling. Say a male frog only croaks once every ten years. You hear a croak and see two frogs. I would think that the likelihood that they are both male increases because for you to have ever heard a croak at all is more likely with two male frogs around then one. Yet only one frog actually croaked.
#31
Old 03-21-2016, 11:18 AM
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OK, I'm going to restate the original boy/girl puzzle, to clarify the issue. You meet a random person, and ask that person whether e has exactly two children. The person says yes. You then ask that person whether it is the case that e has at least one boy. The person says yes. Given the answers to those two questions, what is the probability that the person has two boys? The answer is 1 in 3.

Please note, here, the way that I phrased this. I did not ask "what is the probability that the other child is a boy", because the phrase "the other child" is meaningless here. You can't say "the one other than the one the parent was thinking of", because if the parent had two boys, e was thinking of both of them when answering the question. And this distinction is crucial, because if the puzzle is ever rephrased in such a way that "the other child" becomes meaningful, then the odds change back to the intuitive 1 in 2. For instance, if I ask the parent "Think of one of your children. Is that child a boy?", then I can refer to "the other child", and the other child is equally likely to be a boy or a girl.

And the way the frog puzzle is phrased, there is indeed meaning to "the other frog": It's the one that didn't croak. You can't tell which one that is by looking at it, but by the same token, you can't tell which child the parent was thinking of. It doesn't matter: "the other frog" is well-defined, and so the odds are 50%.

Quote:
Quoth Biotop:

Why would you assume a silent frog is equally likely to be male as female? If a male sometimes croaks and the female never croaks, and male frogs and female frogs are evenly distributed, then a non-croaking frog is more likely to be female.
This is true. We don't know how much more likely, since we don't know how often male frogs croak, but a non-croaking frog has some chance greater than 50% of being female. But this doesn't change the puzzle: Each direction has exactly one non-croaking frog, and so whatever the chance is of a non-croaking frog being female, it's the same in both directions.
#32
Old 03-21-2016, 11:21 AM
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Originally Posted by Biotop View Post
This is where I am still struggling. Say a male frog only croaks once every ten years. You hear a croak and see two frogs. I would think that the likelihood that they are both male increases because for you to have ever heard a croak at all is more likely with two male frogs around then one. Yet only one frog actually croaked.
Yes, playing around with Bayes' theorem (and hoping that I haven't messed it up), it seems that if you assume any particular probability of a random male frog croaking in a particular period, the period in this case being the time our protagonist has to decide, then it doesn't in fact make any difference whether you choose the lone frog or the pair of frogs. The chance of getting a female is the same.

If we select two frogs randomly and observe them over some period, then

p(both frogs are male, given that exactly one of the frogs croaks during the period)

= p(exactly one of two randomly chosen frogs croaks during the period, given that both frogs are male) x p(both frogs are male)
divided by
p(exactly one of two randomly chosen frogs croaks during the period)

And

p(a frog is female, given that it didn't croak during the period)

= p(randomly chosen frog didn't croak during the period, given that it is female) x p(a randomly chosen frog is female)
divided by
p(randomly chosen frog didn't croak during the period)

= 1 x 1/2 / p(randomly chosen frog didn't croak during the period)
= 1 / (2 x p(randomly chosen frog didn't croak during the period))

If we assume that a male frog has a 0.5 probability of croaking during the period (so a random frog has a 0.25 probability, because it could be female), then

p(both male | exactly one of the frogs croaks during the period) =
(2 x 0.5 x (1 - 0.5)) x 0.25 / (2 x 0.25 x (1 - 0.25)) = 0.333... . So the chance of getting a female frog in the pair is ~67%.

But the chance of the lone, non-croaking frog being female is
1 / 2(1 - 0.25) ~= 67%. It's the same. That is, there's no reason to prefer the pair of frogs to the lone frog.

If male frogs croak a lot, say p = 0.9, then naturally we suspect that a non-croaker is female, and the calculation bears that out:

p = (2 x 0.9 x (1 - 0.9)) x 0.25 / (2 x 0.45 x (1 - 0.45)) = 0.0909...

That is, they're much less likely to both be male. So about a 91% chance of getting a female frog. But the chance of the lone frog being female is

1 / 2(1 - 0.45) = ~91%

Again, it's the same. The answers are always the same, whatever probability to you choose for a male frog croaking. So there is no reason to prefer the pair of frogs to the lone frog.
#33
Old 03-21-2016, 11:34 AM
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Now, if I'd seen Chronos's post in time, I wouldn't have had to do all that calculation
#34
Old 03-21-2016, 11:59 AM
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I think I get it now. If the riddler had said that it is known that two female frogs never travel together in pairs-- but they otherwise comingle, then the odds would be better to lick the two.

But if we actually have a frog proclaiming, "I am male," then the non-speaking frog's femininity odds are 50%.
#35
Old 03-21-2016, 12:13 PM
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Wait a minute...

Grr...

Suppose the person first looked to the left and saw two frogs which could be either sex, but then looked to the right and saw one frog. While he was looking to the right, he heard one of the two frogs on the left croak.

Yes you have exactly one non-speaking frog in each direction. But you do not know which is the non-speaking frog on the left. All you can eliminate is the f/f possibility.
#36
Old 03-21-2016, 12:28 PM
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Ok. I think I finally see now. While we don't know which frog croaked, we do know that a specific frog did croak. We therefore we can also eliminate the chances where a frog that was not the specific frog croaked.

This puzzle makes me want to croak.

Last edited by Biotop; 03-21-2016 at 12:28 PM.
#37
Old 03-21-2016, 12:52 PM
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Join Date: Mar 2000
Location: Faber, VA
Posts: 6,470
Sigh. Can someone smarter than me tell me where I am mistaken on post #22? Because I think I understand and then lose it again.
#38
Old 03-21-2016, 12:57 PM
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Join Date: Mar 2000
Location: Faber, VA
Posts: 6,470
DING! Light bulb goes off!

Post#22 is correct (ignoring the 75% stupidity at the end), but just saying the same thing... that it doesn't matter which way you go.

Last edited by Biotop; 03-21-2016 at 12:58 PM.
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