I saw this video recently:

https://www.youtube.com/watch?v=cpwSGsb-rTs

Basically, you are poisoned in the jungle and the antidote is on the skin of the female of a certain species of frog. You are about to die but 50 feet away you see one of these frogs. 50 feet the other way, you see two of them. You can only go 50 feet (either to the one frog or the pair or two) and lick them before you die.

The male frogs croak a certain way, and you hear that from the direction of the pair. The puzzle asks, where are you odds of surviving better? Running to the pair or to the single frog?

The video says the answer is the pair of frogs. The explanation is that the single frog has a 1-in-2 chance of being female, but there is a 2-in-3 chance that one of the pair is a female. That is, before the croak we can presume 4 possibilities (MF, FM, MM, and FF) but because a male croak was heard we can strike the FF possibility.

The problem I have with this explanation is that we are not looking for any pair of frogs but a single frog. We need the female. Why should it matter that one of the two is a male? The other has a 50-50 chance of NOT being a male. I am specifically reminded of gambler's fallacy here.

Put another way, the chance of any frog being male or female is 1-in-2. If we randomly select frogs one at a time, there is a 50% chance each time of either gender. We could alternately select 2 of them first and check each one's gender after. Upon inspecting each one, we should have the same chance of having a male or female each time. It shouldn't matter when any particular frog was drawn from the pool.

If, on the other hand, I needed both a male and a female to survive (but not male-male or female-female), then I would agree with the video's conclusion.

Can someone help me understand this puzzle?

I'd also say it's 50-50. if there are 2 frogs and 1 is definitely male, still we have 1 unknown frog which gender can be either male or female with 50-50 chance, and we only need a female.

there are conditional probability problems that can be unintuitive at first (check monty hall problem if you want), but I think they messed it up in the video.

Do you only get to lick one frog?

You get to lick both frogs if you choose the pair.

Then it's obviously 2 out of three. But it's kind of a dumb problem because if you had no information that one of the two frogs was male it would be 3 out of four.

There are two frogs - frog A and frog B. Which one is the male?

It's 50-50 here. They're trying to get https://en.wikipedia.org/wiki/Boy_or_Girl_paradox but ****ing up the information gathering procedure necessary to achieve it.

There are 3 states left once you hear the male croak- MM, MF, FM, that were initially equally likely, but given that you could have heard a female croak (if one were there), but didn't, the MM state is twice as likely relatively- by Bayes:

P(MM|M croak) = P(Mcroak |MM)*P(MM)/P(M croak) = 1 * .25 / .5 = .5.

P(MF|M croak) = P(Mcroak |MF)*P(MF)/P(M croak) = .5 * .25 / .5 = .25

P(FM) = P(MF) = .25

So you're at 50% MM, 50% MF/FM. So 50% for a second male, 50% for a female. To get the paradox, you have to learn the information that a male is in the pair WITHOUT doing it in a way that makes it more likely for you to learn that from a MM pair than a MF pair- which clearly isn't the case here because the probability of having heard a M croak is double in MM relative to MF/FM.

If you said a quick prayer and asked if at least one of the pair was a male, and god told you yes, then you'd have the paradox.

Thanks for your input. Others critical of the video have also zeroed in on the vague problem description.

I will have to study your Bayes equasion. I do want to point out that the females do not croak, but not sure if that matters. The males MAY croak.

According to one rebuttal, the combos after 1 croak should be extrapolated as follows: M'F, FM', M'M, and MM' where M' is the male that croaked. In this case we would still arrive at 50% for a female in the pair.

I don't believe it matters. We know one is a male, but the other is unidentified. And I don't see any interdependence - unlike the Monty Hall problem.

Oh, that formulation is total trash then. It's obviously not even 50-50 that the singleton is a female because every instant that you don't hear a croak from over there makes female more likely. Don't even waste time on this version of the puzzle.

Both are unidentified. But between the two, at least one is male.

Thanks for clearing up a source of confusion for me! Your statement explains why we have to consider the pair as a whole, not just "the one which didn't croak" (since we don't know which one it is).

If we somehow magically know for certain that one of the pair is a male, then the combos are MF, FM, and MM and they are equally weighted, so the chance is 2-to-1 that there is a female in the pair. This is probably what the creators of the video were going for.

But in their problem description we should take croaking into account. The combos are no longer equally weighted. We don't know how to weight the MM combo because we don't know the frequency at which males croak, but if we assume a 1-in-2 chance by the time of our decision, we can break MM out into M'M and MM' and now we have 4 equally weighted combos, 2 of which contain a female.

Sound reasonable?

Monte Hall redux.

The only problem is they confused the problem with the issue of how often males croak. We must assume the creator just wanted it to be a given (clue from Monte) that one frog is male (one door is a goat). And that makes the two problems nearly identical logically.

As a former teacher, this puzzle is frustrating to see. Not only are there logical probabilistic errors in the formulation as Tom and others have pointed out but the conceptual basis is faulty. AFAIK frog skin is not antidote to anything and some frog skin is even poisonous. So licking frog skin it is already unnatural. Also, in nature, male and female frogs are distinctly different in size so there would be easy to differentiate (I'm aware the puzzle makers tried to patch this flaw) . There are better scenarios to choose from.

Solarglow, no, to make this puzzle work properly, the poisoned person would have to know the croaking frequency rate of both male and females at the very least. To simplify the puzzle, maybe you could assume the poisoned person knows that all frogs, male or female, each croak rarely, maybe once a week, and the poisoned person needs to find an antidote within a few minutes. Male croaks must sound different from female croaks. Best to get rid of the frogs altogether and compose a new puzzle from scratch.

agreed, its just monty hall

Don't think it is Monty Hall as the point of that puzzle is that the new information and changing your choice improves your odds of being correct.

In this case the information reduces the odds of being correct. You would always intuitively go to the two frogs if you heard no croaking (cos 2 is better than 1 amirite?) and as MM are only 25% of the possibilities, you would obviously be correct. Hearing the male croak deletes the possibility of FF, so you are are now a 33% chance to get MM.

OK it makes a point about that despite the fact you have two unknown frogs to choose from, one choice is superior to the other, but if I had to rate it as a puzzle I would give it 1.5 stars.