# 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: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: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.

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.

Unless males only croak in the presence of females.

It’s really just a restatement of the Boy/Girl Paradox.

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.

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.

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.

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.

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.

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.

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.

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.

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?

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.

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.

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?

This. Well explained.

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.

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.

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.

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.