# TED-Ed’s Frog Riddle Is Wrong

TED-Ed presented a riddle last week based on a classic probability problem. However in the riddle there is a small and seemingly insignificant detail that changes the calculation. In this video I present the pertinent details of the frog riddle, explain its connection to the boy or girl paradox, and then do a detailed calculation of what I believe is the correct probability.

TED-ED frog riddle: https://www.youtube.com/watch?v=cpwSGsb-rTs

Blog post (another calculation if the probability a male frog croaks is p): http://wp.me/p6aMk-4wD

Ron Niles made a video that shows the probability visually and explains an interpretation of a male frog croaking with probability p: https://www.youtube.com/watch?v=K53P544P1vE

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### 32 thoughts on “TED-Ed’s Frog Riddle Is Wrong”

• September 18, 2017 at 6:51 am

Presh please explain i dont understand. If you dont use any math and just think about it logically, how does it make sense that just because he was born on tuesday, it changes probability from a boy being born?? Who cares if he's born on tuesday? Event a could also be on tuesday. It could be wednesday, too! so?!!?!? Why does the day of the week matter? The boy could be born before the girl. It could be born after it. He can also have 2 boys. same with Event B. the boy could be born before, or after the girl. If the boy was born on tuesday, another boy could be born last sunday. Or a girl. order does not matter!!!

• September 18, 2017 at 6:51 am

the answer is bogus, It doesn't matter if you heard the croak or you know one is male, its the same thing. Furthermore if you know 100% 1 is male that means there is a 50/50 the other could be female.
your sample space should have looked like :
M100% (M?)
M100% (F?)
Therefor 50%

Your further explanations would only be valid if I said here are 2 frogs 1 is 100% male, If you were to choose a frog and guess its gender what would the chance of you picking a female be? The question doesn't ask which one, it doesn't even care.
MM
MF
FM
Now if the question asked what is the chance the left one is female? 1/3. What is the chance the right 1 is female? 1/3.
again the question does not care which frog, only if there is one.

• September 18, 2017 at 6:51 am

The wording for the girl-boy and boy born on Tuesday has its downfall in the wording, in my opinion. The fact that you know 'a boy' in the family was born on a tuesday conveys no useful information in my opinion.

How about this thought experiment? – I've got a massive population of random two child pairs, and I'm going to select one pair at random, and if it's not a girl girl pair, I'm going to tell you the birth-day of 'a boy' in the family – so what you have to do is predict if the other child is a boy – probability ~0.5 or 1/3? What are you going to do if I offer you odds on of 40%? I'm giving you odds of 3 to 2 there – very good if the true probability is around 50%, but terrible if it is 1/3 – would you take those odds, if I told you the selected pair had a boy with a named weekday he was born on? Has it really told you something? Are you sure you're going to make money?

• September 18, 2017 at 6:51 am

A side note here. For those who downvoted this video and those who really can't accept the difference between "given a frog is male" and "hear a male frog sound", try to think the following way: P(hear a male frog sound) + P(hear a female frog sound) = 1, this is obvious because you must either hear a male sound or a female sound. If you are a jerk and say "what about the case where no sound is heard?" It's fine as well, because the sum will be less than one then. But P(one of the frog is male) + P(one of the frog is female) is larger than one , namely 3/4 + 3/4 = 3/2. Think this way and you will find yourself easier to accept the fact that there is indeed a difference.

• September 18, 2017 at 6:51 am

What's the difference between BG (boy/girl) and GB (girl/boy), why are you treating it separately ? We don't/need information which one is older, so this should be one and same. He has 2-elemental set "Children" which has an element "boy" and element "girl". That's it.

• September 18, 2017 at 6:51 am

Almost used the same logic on one of talwakers non bs videos, but then realized I was being an idiot
BG=GB here as well
50% people

• September 18, 2017 at 6:51 am

Mmmm. Confused logic going on here. Are you calculating probability of having a girl, or the probability of there being a girl in the group of two children? The frog situation is different, and depends upon an interpretation, and a definition that samples are different – which they aren't. So I'd argue that the probability, as asked remains at 2/3. I'm not gonna look at anymore of these math riddle/problem videos – reminds me too much of school, where the teacher was often wrong and used to ridicule or even cane students for pointing it out – scarred for life!

• September 18, 2017 at 6:51 am

The problem is in seeing that because one of possible outcomes (FF) was removed, therefore there are only three possible outcomes and two have them have an F.

But what also changes is that the three remaining outcomes do not have the same probability of happening. MM will happen 50% of the time, MF 25% and FM 25%.

Just like a roulette table has three colours in red, black and green. It doesn't mean green has an equal chance as red or black.

• September 18, 2017 at 6:51 am

I actually addressed exactly this issue in a (publicly visible) FB note. It points out exactly how the video was wrong, but only because it actually assumes a specific variable. The 2/3rds probably it gives for there being a female is actually correct, even when considering croaking vs silent males (because of what someone else pointed out, that when you split the columns into two to track silent vs croaking males when there are two males, you must also subdivide the probability of there being two males to begin with). THe reason it's "wrong" is precisely because it very carefully points out the probabilities for everything involved except for the chance that a from would have croaked. If that chance is exactly 50% (and thus completely randomly obtained information), then their numbers are correct. If it's anything else, they're wrong.

Here's the link if anyone wants to take a look: https://www.facebook.com/notes/richard-homan/chance-a-frog-will-save-your-life/10153269302586829/

• September 18, 2017 at 6:51 am

at 8:00, what about the probability that both males croak?

• September 18, 2017 at 6:51 am

Me with the boy/girl problem he presented (event a): but girl and boy and boy and girl are the same thing, therefore it should me 1/2.

• September 18, 2017 at 6:51 am

a boy and a girl or a girl and a boy are the same event and should be concidered as one…

• September 18, 2017 at 6:51 am

It depends on the likelihood that a male frog will croak. The less likely that a male frog will croak the closer it is to 50% and the more likely it is that it will croak the closer it is to 2/3. So if you know that a male frog croaks every second (but you can't know if it is one that's croaking or both) then it is 2/3 chance that there is a female there.

• September 18, 2017 at 6:51 am

Nope. The boy-girl bit in your video was wrong too. You calculated for a sample space that takes into account the relative ages of the children, but your question has no indication this should be accounted for. A similar issue came up with the 2/3 frog calculation as well.

The rest seemed fine though. Within these samples, it should be 50% or very close to it, as are intuition would tell us. There is a reason your intuition is as it is. I appreciate you doing these videos, its good to see someone trying to dispel common misunderstandings.

• September 18, 2017 at 6:51 am

Both variations are incorrect… MF / FM are the same thing, and should not be considered extra events. in all cases if you eliminate equivalent events then the probability there is a girl is 50%

EDIT: Why don't you throw is the possibility that BOTH frogs croaked?! then it's less than 50% likely there is a female frog. The entire methodology for solving this problem is flawed.

• September 18, 2017 at 6:51 am

that like-dislike ratio doh

• September 18, 2017 at 6:51 am

I'm not convinced with this demonstration. The example of boy and girl is not similar to frog example. The croak of frog in the riddle is used to indicate that at least one of the two frogs is male and then both cases 1 and 2 are similar. However for boy and girl example the two cases are different and thus probabilities should be different an i agree with the values he reachs.

• September 18, 2017 at 6:51 am

Yeah I agree, I saw that while I was working out the probability while I was solving the riddle. I saw that there was 50/50 chance the man was going to die since there's an equal number of frogs in both genders but assuming that at least one is female is in the group of 3 then yeah going for the two frogs is the better choice

• September 18, 2017 at 6:51 am

Whether it was the first or the second male frog that croaked, it doesn't really matter because neither of the male frogs have the antidote.

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• September 18, 2017 at 6:51 am

Therefore, half of male frogs are gay ? There are too many dpdgy assumptions.

• September 18, 2017 at 6:51 am

still don't get it why isn't BG and GB the same?? How is it 2/3 chance?? If you have a child that is either boy or a girl shouldn't it just always be 50% probability it's a girl and 50% probability it's a boy.

• September 18, 2017 at 6:51 am

Why are you not considering both frogs croaking in the sample space? Yes, according to the riddle only a male croak is heard. But if you are putting female croaks in the sample space, I don;t understand why you would not put both frogs croaking too. It is certainly a possible event, whether it is the one that happened or not.

• September 18, 2017 at 6:51 am

my intuition says 1/2 so I setup to prove it by using Bayes rule and got the following:
p(1g/1b)=p(1g^1b)/p(1b)
we need p(1b)
p(1b)=p(1b/1b^1g).p(1b^1g)+p(1b/2b).p(2b)+p(1b/2g).p(2g)=1×1/2+1×1/4+0×1/4=3/4
so p(b)=3/4
and from above
p(1g/1b)=p(1g^1b)/p(1b)=(1/2)/(3/4)=2/3
I'm puzzeled by the answer :/

• September 18, 2017 at 6:51 am

please stop complicating easy problems, the calculation you did should come into the picture if and only if the problem said
1) he has a boy and you know one of his children was born on a Tuesday.
OR
2 ) WHAT IS THE PROBABILITY HE HAS A GIRL and one of them was born on a tuesday if you know he has a boy.
This is why people find math scary because people complicate it.

• September 18, 2017 at 6:51 am

Here's a small python script that proves that the the probability in the boy/girl/tuesday problem is indeed 52%: https://pastebin.com/Tj73XHVS
In 100M runs, there were 13.78M pairings with a boy born on tuesday, and in 51.9% of those, the other sibling was a girl.

• September 18, 2017 at 6:51 am

the Ted Talk didn't make a mistake you're expanding the problem with irrelevant information and assuming the chance outcome probabilities of those events that are established to be irrelevant.. for example what if the probability that if there is at least one frog than one frog will croak is 100% meaning that if there are two male frogs the probability that one frog will croak is

• September 18, 2017 at 6:51 am

It's interesting to consider a group of three people walking in the woods. Alice, Bob, and Cindy. Bob is wearing headphones, and Cindy is deaf.

All three eat a poison mushroom and are searching for frogs. All three stumble across the frog on the stump.

Alice hears a frog croaking behind her, and sees the two frogs in the clearing. Alice then gets the attention of Bob and Cindy. Bob takes off his headphones to hear Alice say, "one of those frogs is male." Cindy, being deaf, doesn't hear anything.

Cindy is only aware that there are two frogs in the clearing. She calculates that she has a 3/4 probability of survival if she goes to the frogs in the clearing and a 1/2 probability survival of going to the frog on the stump. So she goes for the clearing.

Bob is only aware that there are two frogs in the clearing and that one of them is male. He calculates that he has a 2/3 probability of survival if he goes to the frogs in the clearing and a 1/2 probability of survival of going to the frog on the stump. So he goes for the clearing.

Alice is aware that there are two frogs in the clearing and that one of them is male because she heard one of them croak . She calculates that she has the same probability of surviving going either way*. Who can say where she goes?

The resolution of this seeming paradox is that the probabilities calculated here do not mean that Cindy is more likely to survive than Bob and Alice. If all go for the clearing, they will either all survive or all die. It's impossible for Cindy to survive but both Bob and Alice to die. The probabilities they calculate are their estimates for survival (by going to the clearing) based on the information they have. If an omniscient being were there too, the omniscient being would conclude that there is either a probability of 0 or a probability of 1 for survival by heading to the clearing (since the omniscient being knows the sexes of the frogs). The omniscient being has more information than anyone, and everyone else is either overestimating or underestimating based on the information they have. The degree to which their estimate is wrong depends on how much (or little) information they have.

*When it comes to Alice's calculation, there are various arguments here, depending on the probability that a male frog croaks. No matter what the probability it is that a male frog croaks, you would calculate that you are equally like to have a female frog in the clearing as you are to have a female frog on the stump – both occur with probability 1/(2−p) where p is the probability that a male frog croaks – thanks to HumptyDumptyOakland for pointing this out in the comments of this video and Presh Talwalkar for doing a computation like this in his blog post (see the video description)! Regardless of what p is, however, both the stump and the clearing have the same probability of having a female, so regardless of the value of p, Alice would not decide one way was better than the other.

• September 18, 2017 at 6:51 am

Boy-Girl problem: You're not answering the same question that you're asking.

You're asking: Given that he has 2 children, and one is a boy (that incidentally was born on a Tuesday), what is the probability that his other child is a girl. Whether the other child is a boy or a girl is unrelated to the fact that the first child (not necessarily first born) was born on a Tuesday. The the fact that the first child was born on a Tuesday is just a premise of the QUESTION, just like the premise that he has 2 children and that one of them is a boy.

But the question being answered is something like: Given that he has 2 children, and one is a boy what is the probability that he was born on a Tuesday and his other child is a girl, or perhaps: Given that he has 2 children, and one is a boy that was born on Tuesday, what is the probability that he was born on a Tuesday and his other child is a girl.

The difference is that in the question, the day of the week that the boy was born on is incidental, … just a part of the conditions of the QUESTION. Just like other irrelevant facts like he was born on an odd or even day or year, or born in May, or born "Last Tuesday", or wants to become President when he grows up.