Can you solve the frog riddle? - Derek Abbott
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You’re stranded in a rainforest, and you’ve eaten a poisonous mushroom. To save your life, you need an antidote excreted by a certain species of frog. Unfortunately, only the female frog produces the antidote. The male and female look identical, but the male frog has a distinctive croak. Derek Abbott shows how to use conditional probability to make sure you lick the right frog and get out alive.
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by Eric Chen
Eric Chen
Lesson in progress
The answer is 50/50 and I will prove below how TED messed up.
The combination set that TED proposes: mm mf fm ff (excluded) is not the true combination set. The problem clearly states that we heard one male frog croak. I will denote this frog with a capital M and any other potentially male frogs with a lower case m. Your combination set is actually this: Mm mM Mf fM ff (excluded) mm(excluded) fm(excluded) mf(excluded) You see, the frog that croaked and proved it was male could have potentially been the first or the second of the pair of frogs, and so both situations give a 50% chance for the other frog to be male or female. The actual probability is thus 50/50. I'd also like to mention that combinations is not even the right way to do this problem.
Comments are closed on this discussion.
Eric Chen
Eric Chen
Lesson in progress
in response to Daniel Nordling Show comment
Order actually does matter here. It's hard for some people to grasp the illusion in this problem, so I'll do my best to explain.
To say that the order does not matter is like saying 8 / 4 and 4 / 2 are the same. They mathematically give you the same answer, but when you bring it into the real world, they are not the same. Think if you have 8 apples and divided it evenly among 4 people, vs 4 apples among 2 people, each person gets the same number of apples but the total number of people and apples is different.
In the same way, saying MF and FM are the same will give you the right answer, but technically in this problem order does matter, and since order does matter, MF and FM are not the same, but the people that are saying this are also missing the other half of the answer. If MF and FM are not the same, then there are also 2 combinations of MM as well as FF. In other words, we can say MM and MM are different depending on which M you picked first.
Daniel Nordling
Lesson in progress
About the Sample space. People keep stating MM, FF, MF, FM. Aren't MF, FM technically the same? The order of the frogs doesn't really matter in this case.
Eric Chen
Eric Chen
Lesson in progress
in response to Antonis Tsirilikos Show comment
Whether or not the frog croaked makes no difference. If you are to assume that we know one is male regardless of croaking, then the one we know is male takes the place of the frog that croaked in the combination set. It yields the same result. People tend not to acknowledge that the one we know is male is at a 100% probability for some reason, but that is an important fact that comes into play.
On the sleeping beauty paradox, the answer is 2/3 tails. The 1/2 answer is absolutely wrong, and I'm not going to bother explaining it here as that's a different discussion..
Antonis Tsirilikos
Lesson in progress
in response to Eric Chen Show comment
Indeed, the boy/girl paradox is a different problem.
There is a problem quite similar to frog problem but easier to understand. The sleeping beauty Dilemma. https://www.quantamagazine.org/20160129-solution-sleeping-beautys-dilemma/
Both these riddles are made this way so it is difficult to comprehend the question. Once you anderstand, the calculation is easy. I think the sleeping beauty Dilemma is easier to understand, so it can help to understand this one too. Still there are people who think the correct answer is 2/3. They are called "thirders", while those who support the 1/2, are called "halfers"!!! LOL!
Antonis Tsirilikos
Lesson in progress
in response to Steven Chanan Show comment
I think this guy is wrong too. The answer should be 50% no matter how the information about male frog is obtained.
Antonis Tsirilikos
Lesson in progress
Exactly!
One thing remains though. You are using the fact that the frog croaked. What if we know there is a male in the pair and that is all we know. No croaking. We only have this information and no other. In this case, will the solution be the same or will give a 2/3 propability?
Eric Chen
Eric Chen
Lesson in progress
in response to Terence Schwarzbrodt Show comment
I understand your point, but the distinction I am trying to make is that the question we are considering in the frog problem is not whether there is a female frog, but rather whether or not the chance of survival is different on either side. When the question is worded that way, it adds the extra point that we do not know, therefore we are forced to treat it as a random sample to give us our odds.
Even Martin Gardner, the person that is credited for making the boy/girl paradox known admits that the question is ambiguous, depending on how the information was found out and how the person was selected. As such, it is not an actual probability problem.
Steven Chanan
Lesson in progress
in response to Terence Schwarzbrodt Show comment
I haven't gotten a chance to jump back into this, but I will mention that there is a thorough discussion here: http://mindyourdecisions.com/blog/2016/03/08/ted-eds-frog-riddle-is-not-entirely-correct-game-theory-tuesdays/. Fwiw, it both agrees with Eric's analysis and compares the problem to the boy/girl paradox.
Terence Schwarzbrodt
Lesson in progress
in response to Eric Chen Show comment
That "he/she is already born" is not exactly the best argument to support that "this [the frog riddle] has nothing to do with the boy/girl paradox", given that the frog is already male or female as well.
Steven is right that this is a variant of the boy/girl paradox and that the distinction you are focussing on is irrelevant, as the problem could easily be reformulated to turn it into a "valid" probability question.
Steven Chanan
Lesson in progress
in response to Eric Chen Show comment
Thank you so much for your kind response. For my part, I'm sorry that I didn't really consider that it might be a language issue.
Honestly, I'm quite fascinated by this conversation and want to continue it further. Gotta run now, but I'll post again soon...
