Basic probability : the frog riddle - what are the chances?

A few days ago I was watching this video The frog riddle and I have been thinking a lot about this riddle.

In this riddle you are poisoned and need to lick a female frog to survive. There are 2 frogs behind you and basically, you have to find what are your chances to find a least one female in these two frogs (you can lick both of them). The only thing is : you know one of them is a male (because your heard the croak) but you don't know witch one.

The video solves the problem with conditional probability and explains that you have a 2/3 chance of getting a female. (on the four possibilities MM / MF / FM / FF, knowing there is a male eliminates FF)

Here is my question : If you see which one is a male (for example the frog on the left is a male) what are your chances to survive ? Is it 1/2 ? because we only have two possibilities (MM or MF) with probability 1/2. Is it still 2/3 because the position does not matter ?

Bonus question : If it is 1/2, then if close your eyes and the frogs can move, is it still 1/2 or does it comes back to 2/3 ?

Similar problem : If I have two children, and I know one is a son, then I have a 67% chance to have a daughter. But if I know the oldest one is a son, then I have a 50% chance to have a daughter. Is it exactly the same problem here ?

Can you please explain this to me ?

• You need to use the search function - this problem has been discussed twice in the past week (on reddit as well). The frog riddle is fundamentally different from the Boy-Girl situation in that MM are more likely to produce a croak (sooner) than FM. If you hear one croak from the back, you are indifferent between front and back. – A.S. Mar 8 '16 at 12:38
• @A.S. could you link to the answer so it can be marked as a duplicate? I can't find it. – Jamie Twells Mar 15 '16 at 13:47

After some reasearches :

Here is an interesting answer on StackExchange explaining why the video is wrong : the croak does not simply give the information that there is a male (because of the craok, MM is more probable than MF or FM)

A reddit question on this topic with a lot of questions and answers about how to handle the problem

And an interesting Wikipedia article on the bertrand paradox explaining why you have to define a problem correctly. I think here the problem is not perfectly defined.

The answer that @Lordofdark linked to is very interesting. In addition to the matter of the probability of croaking, there are other factors to consider. For instance, the problem doesn't specify whether the sex of the frogs is independent. For all we know, male frogs might never be found alone, so we should definitely lick the solitary frog. Or perhaps it's mating season and pairs of frogs are always male/female. Perhaps male frogs only croak when they're around females. Or...

Simply put, there is not enough information to go on. A mathematician lost in the jungle will probably die from taking too long to consider the possibilities.

Since you can lick both frogs the order in which we place the frogs are irrelevant. there are only two possibilities FM and MM. FF being eliminated. The chances are 50%. Knowing which one is the male, saves you one lick but the probability for the other one being a female is still 50%. (Same situation for the boy-girl problem)

It depends on the probability of croaking. If we assume that there are an equal number of croaking males (C) and silent males (S), then we can draw the sample space (females being F):

CS SC CF CF FC FC

Where CF and FC are repeated twice as there are twice as many females than croaking males or silent males. Thus, assuming the probability of a frog croaking is 1/2, the probability of there being a female is 2/3.

The answer is 2/3 because on the clearing you hear a croak and see two frogs and only the female has the antidote. There are 4 possible combination of frogs. Male and female, female and male, female and female, male and male. Since you heard the croak, the combination female and female is out which leaves 3 combinations one which will kill you. So out of those 3 combinations 2 of 3 will help you survive which is 2/3 or about 67% of survival.

protected by Alex M.May 26 '17 at 7:10

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