The frog riddle

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The frog riddle

Whats your take on this puzzle?
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Qaanol
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Re: The frog riddle

There is not enough information provided to solve the puzzle. We would need a model for the social behavior of frogs, to determine the probability that a pair of frogs found together would have each possible combination of genders, as well as the probability that a lone frog would be male or female.
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Eebster the Great
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Re: The frog riddle

As a probability puzzle, it is pretty straightforward and the narrator does a decent job of explaining it. Qaanol, the narrator says that each frog's sex is independent.

gmalivuk
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Re: The frog riddle

Whats your take on this puzzle?

I don't understand what or why you're asking. The video explains how the puzzle works.

Do you disagree with the explanation? Do you want someone here to help you understand it? Do you just want to know whether we think it's a good puzzle?
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Tub
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Re: The frog riddle

*sigh* that's not how conditional probability works.

Just like the endless misunderstandings of the monty hall problem, this one depends on a hidden question: What's making the frog croak?
You gained some information, but how did you gain it, and how likely were you to gain exactly this information in each possible starting configuration?

The monty hall problem only works if the host has knowledge of the doors, and is bound by the rules of the game show to always reveal one door with a goat. Say you pick door A, and the host reveals door B. The chance for revealing door B when the price is behind door B is 0%, when it's behind door C it's 100%, and when it's behind door A it's 50%. Do the math, then pick door C.
The information "B is a goat" will only ever increase your odds to 1/2. Two doors left, same chance for each. The crucial information is "the host chose to reveal door B", which has different likelihoods in different starting configurations.

Same here. We've established that the frog's genders are independent for this riddle. If you pick one of the two frogs and you tickle it until it croaks, then that croak is not going to tell you anything at all about the gender of the other. That's the definition of independent events.

If by decree of the wood fairy exactly one of the two frogs is required to reveal its gender during the riddle, then you need to include that event in your conditional probability. Realize that hearing a male gender reveal is more likely when there are two males, and when weighting the configurations accordingly, your conditional probability P(one frog is female | one frog is revealed as male) is exactly 0.5. Again, no information about a single frog will tell you anything about the other.

If there's a game show host who has knowledge of *both* frogs, and who's required to tell you either "there's at least one male frog" or "they're both female", then after hearing the first statement you can correctly expect a 2/3 chance for a female.
(Remember: this is bad news, because two frogs otherwise have a 75% chance for one female, and now you're down to 66%.)

tl;dr: the video refuses to give a crucial piece of information. The given answer is thus neither correct nor false; it's meaningless.

gmalivuk
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Re: The frog riddle

But then by your account conditional probability is never usable because there's always additional unknown information that would further change the likelihood of different things happening.

The point the video shows, correctly, is that the knowledge that at least one is male (not that a particular one is male, which is what your tickling scenario would show) should change your estimate of the likelihood that they're both male.
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Tub
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Re: The frog riddle

gmalivuk wrote:But then by your account conditional probability is never usable[...]

So the idea that I cannot account for every small factor means that I get to ignore the big factors without having to care about confidence or error bars?

gmalivuk wrote:The point the video shows, correctly, is that the knowledge that at least one is male (not that a particular one is male, which is what your tickling scenario would show) should change your estimate of the likelihood that they're both male.

The information you're getting is not "at least one of them is male" (which I agree yields 2/3, per my third scenario), but "one of them croaked", which allows for different valid interpretations. In a set of independent variables, revealing one variable cannot give you information about the others, no matter if you reveal a specific variable or you pick a variable to reveal at random. And, as worded, this sounds a lot like revealing a single variable.

Both Monty Hall and Betrand's paradox teach us to be precise about these things, because they matter. If the riddle can't be precise about them, the riddle doesn't get to claim a precise answer. It certainly doesn't get to claim that 50% is obviously wrong.

This could have been avoided with a proper setup, for example: "You know that any time two female frogs meet, they both start croaking. These two are silent." Now it's knowledge of the two-frog system and not of a single frog, 2/3 is the only correct answer, good riddle.

Eebster the Great
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Re: The frog riddle

I will grant that from a practical standpoint, the problem doesn't really work. If male frogs always croak, then we already know the non-croaking frogs are female. If male frogs sometimes croak and sometimes don't, then a pair of males is more likely to produce at least one croak in the given interval than a single male. Croaking frogs are not the ideal setup.

However, we can make a pretty minor adjustment so it works. Instead of croaking, males reveal their presence through a distinctive scent. Unfortunately, there is no way for the human nose to reliably tell the difference between the scent of just one male and of two males. Moreover, the wind is blowing such that you can only pick up smells from the direction of the clearing, not the stump. Now in this scenario, still assuming normal stuff like that the sexes are all independent, the problem should work properly.

The thing is, the video is obviously not about frogs. It's about conditional probability. And its description of how conditional probability works here is correct.

PM 2Ring
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Re: The frog riddle

I didn't watch the video, so I'm not 100% certain of the details of this frog riddle, but it sounds like it's closely related to the Boy or Girl Paradox.

Flumble
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Re: The frog riddle

Yes, it is. The video's scenario makes it a bit harder to accept the answer because of the reasons given so far.

Surely the frog croaks loudly because it's calling for a female, so the one next to it must be another male.

Derek
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Re: The frog riddle

Males of many species are also known to make loud calls after successfully mating though. Exhibit A.

gmalivuk
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Re: The frog riddle

Tub wrote:
gmalivuk wrote:But then by your account conditional probability is never usable[...]

So the idea that I cannot account for every small factor means that I get to ignore the big factors without having to care about confidence or error bars?

Then call it confidence rather than probability if you like. There's a sense in which that's already what we're talking about whenever the thing under consideration has already happened.

These nitpicky objections always come up in conditional probability but rarely in "simple" probability, which makes them tend to feel disingenuous.

If you see one frog to your right and two to your left, which direction is more likely to have the female that saves you? For some reason almost no one ever objects to the claim that the pair is the safer bet even though you're still missing information that could totally change the relative odds. Maybe these are Alex Jones's frogs and all the males have been turned gay, so males all go around in pairs and females are mateless and solitary.
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Qaanol
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Re: The frog riddle

Eebster the Great wrote:As a probability puzzle, it is pretty straightforward and the narrator does a decent job of explaining it. Qaanol, the narrator says that each frog's sex is independent.

No, the narrator’s statement of the puzzle does not say that. It does say that male and female frogs occur in equal numbers, but it does not make any claim about independence.

The only time anything is mentioned as independent is during one of the examples of an incorrect answer, which is not part of the statement of the puzzle.

And it certainly matters, since one could easily imagine a species where individuals always live alone except when they pair up to mate, in which case the two frogs together would definitely include a female.
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Eebster the Great
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Re: The frog riddle

It looks like you're right, he mentions it later in the video. I think the fact that it's an idealized problem is pretty clear from the outset though, even if not every assumption is stated explicitly.

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Re: The frog riddle

This is so wrong
Your 67 percent chance is taken from there being two options of male and female
This isn't true
You have one chance of getting male and female
It doesn't matter the order they are in
You actually have a 50% 50% on each side

SecondTalon
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Re: The frog riddle

Tub wrote:If there's a game show host who has knowledge of *both* frogs, and who's required to tell you either "there's at least one male frog" or "they're both female", then after hearing the first statement you can correctly expect a 2/3 chance for a female.
(Remember: this is bad news, because two frogs otherwise have a 75% chance for one female, and now you're down to 66%.)

tl;dr: the video refuses to give a crucial piece of information. The given answer is thus neither correct nor false; it's meaningless.

We have that information. We hear the male croak. There are two frogs. There are four possible gender configurations with a pair of frogs, the croak eliminates the all female one.

We’re left with 2/3 odds that at least one of them is female, because each frog gender is independent.
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gmalivuk
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Re: The frog riddle

This is so wrong
Your 67 percent chance is taken from there being two options of male and female
This isn't true
You have one chance of getting male and female
It doesn't matter the order they are in
You actually have a 50% 50% on each side

The order doesn't matter to you, but you still have to keep track for counting correctly.

For example, when you look at the frogs, there's one more to the left and one more to the right (if they're directly in line, count the front one on the left). There's a 25% chance the one on the left is male and the one on the right is female, and there's a 25% chance that the one on the left is female and the one on the right is male. There's also a 25% chance that they're both males, and a 25% chance that they're both females.

There are four possibilities for two (ordered) frogs, and then we ignore the order when we're just looking for combinations, so we lump the different-sex pairs together as a single option. But we only count correctly when we understand that this single option has a higher probability of happening than either of the two same-sex options.

Edit: If you don't care about order, and never want to acknowledge any order at any point in your figuring, you still need to understand that some options are more likely than others.
Suppose you had a random selection of 10 frogs. You only care about three options: all males, some mix of male(s) and female(s), all females.
It should be fairly easy to see that all males is a 1/1024 chance, and the same for all females (assuming independent 50% chances of each).
This leaves 1022/1024 (99.8%) for some mix. There are no other options for binary-sexed frogs (which the ones in the thought experiment are assumed to be), and whether or not we care about the order of males and females in the mixed case isn't magically going to make all-male or all-female groupings each 33% likely.

The same logic applies here with just two frogs: all-male = 1/4, all-female = 1/4, mix = 2/4 = 1/2. Then one frog croaks and tells us we're not in the all-female case, and out of the remaining options we have a 1/3 chance of all-male and a 2/3 chance of a mix.
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Eebster the Great
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Re: The frog riddle

Similarly, if I flip a coin twice, I know that I am twice as likely to get a heads and a tails as I am to get two heads. And if I know for a fact that I will not flip two tails, I now have a 2/3 chance of flipping a heads and a tails and a 1/3 chance of flipping two heads.

SuicideJunkie
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Re: The frog riddle

Eebster the Great wrote:Similarly, if I flip a coin twice, I know that I am twice as likely to get a heads and a tails as I am to get two heads. And if I know for a fact that I will not flip two tails, I now have a 2/3 chance of flipping a heads and a tails and a 1/3 chance of flipping two heads.

If flipping the same coin twice, the probabilities can't change between flips. So if the experiment can't result in two tails, then two heads is guaranteed and your coin is very unfair.

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Re: The frog riddle

SuicideJunkie wrote:
Eebster the Great wrote:Similarly, if I flip a coin twice, I know that I am twice as likely to get a heads and a tails as I am to get two heads. And if I know for a fact that I will not flip two tails, I now have a 2/3 chance of flipping a heads and a tails and a 1/3 chance of flipping two heads.

If flipping the same coin twice, the probabilities can't change between flips. So if the experiment can't result in two tails, then two heads is guaranteed and your coin is very unfair.

The idea is that you flip two fair coins (or a fair coin twice) with your eyes closed, or maybe your friend flips them for you behind a screen or something. Then your friend tells you whether or not they are both tails. The rule is that your friend always tells you that and only that piece of information (and is always honest), and you know that rule (and trust it absolutely).

Given that your friend has told you the coins are not both tails, what is the probability that they are both heads? 1 in 3, as stated. Before you got the info from your friend, it was 1 in 4, but now you know better. This is the correct method to adjust probabilities when you learn new information.

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Re: The frog riddle

The difference between knowing after the flips, or before the flips.

Eebster the Great
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Re: The frog riddle

SuicideJunkie wrote:The difference between knowing after the flips, or before the flips.

You missed the super obvious context that I knew I wouldn't flip two tails because of information I got from a time traveler from the future.

SuicideJunkie
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Re: The frog riddle

In that case you can't trust the info; the traveler came from a future where you didn't flip two tails, but the travel will influence the results this time.

Although with enough experimentation with more sensitive quantum effects, you could be reasonably confident whether or not the universe ensures self-consistency in time travel.

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Re: The frog riddle

SuicideJunkie wrote:In that case you can't trust the info; the traveler came from a future where you didn't flip two tails, but the travel will influence the results this time.

Although with enough experimentation with more sensitive quantum effects, you could be reasonably confident whether or not the universe ensures self-consistency in time travel.

You gotta assume the Novikov self-consistency principle. If we're allowing branching, then we have to worry about whether it's like Back to the Future time travel, Groundhog Day time travel, Terminator time travel, or what. Maybe it's Star Trek time travel and too much change can destabilize the timeline and destroy the universe. It's just too much to think about.

Really, these are elementary assumptions in probability riddles involving frogs, dice, and time travel.