Hi, this is Presh Talwalkar.
Last week, the TED-Ed youtube channel, presented the frog-riddle,
which is a variation of a classic probability problem.
However, in telling this story,
they included a slight detail, which would actually change the calculation.
In this video, I'm going to present, what I consider to be, the correct calculation.
I'm going to go over a simplified version of the riddle, with pertinent details.
I'm going to explain how it relates to a classic probability problem,
and one of its variations.
And I'm going to show why the TED-Ed video, actually meant to do one calculation.
But they presented the story for another calculation.
So here's the riddle.
Imagine you have a population of frogs that are equally likely to be male and female, and you cannot visually tell the difference between the genders.
You see two frogs.
what is the probability that there is a female in the pair of frogs, under two different circumstances:
Circumstance A: The pair of frogs contains a male frog.
This is the calculation the TED-Ed video wanted to present.
What the story actually presented, was variation B
The pair contains a male frog because you heard a croak that you know is distinctive to male frogs.
It might seem to you, that variations A and B, are going to be the same.
In either case you knows there's one male frog in the pair.
But the detail, that you heard the croak of the frog, actually changes the probability.
So, in order to explain why, I'm going to go over the classic probability problem this is based on.
The Boy or Girl Paradox.
Imagine Mr. Jones has two children.
You want to know the probability that he has one girl, under two different variations.
Variation A: He has a boy.
Variation B: He has a boy born on Tuesday.
It seems like these two variations would have the same probability.
But let's go over the calculation to see why it's actually different between A and B.
So variation A:
Lets consider Mr. Jones having two children.
What's the sample space?
We know that he could have two boys.
We know that he could have a boy and a girl.
He could have a girl and a boy.
Or he could have two girls.
These are four events, and they're equally likely to happen,
if we assume the probability of having a boy and a girl is equal.
If we know that he has a boy,
the event, that he has two girls, is not possible.
So we have a new sample space,
in which there are three different events: he has two boys, a boy and a girl, or a girl and a boy.
And these are equally likely events.
What is the probability that he has a girl in this new sample space?
The events: boy/girls, and girl/boy are equally likely.
There're two events in which he has a girl out of three.
Therefore the probability, he has a girl, if we know he has a boy, is equal to two thirds.
Now let's consider the other variation
What would happen if we know he has a boy that's born on a Tuesday.
Now this seems like it shouldn't affect the calculation.
Knowing he has a boy, obviously the boy would have to born on some day of the week.
So whether he is born on Tuesday or any other day shouldn't matter.
However the fact we learn, that the boy is born on a Tuesday, does change our sample space.
So we know there're are only three possible pairs of genders, it could either be boy/boy, boy/girl, or girl/boy.
He can't have two girls, because we know that he has a boy.
We also know, that the boy's born on a Tuesday.
So lets indicate the days of the week by a numerical code:
Zero equalling Sunday,
one equalling Monday,
and counting all the way up until six equalling Saturday.
We'll write the day of the week that a child is born, using a subscript.
So what is our sample space now?
For boys that are—boys and girls that are born on different days of the week.
We need to define the events that are equally likely to happen.
If he has two boys, it's possible that the first child was born on a Tuesday,
and the second child was equally likely to be born on any of the other—any of the seven days of the week.
It could also be that the older child is born on a Tuesday, and the f—
—the other child is born of any of days of the week, equally likely to happen.
If he has a boy/girl, we know the boy has to be born on a Tuesday
and the girl can be born on any days of the week.
And similarly for our last case.
There's one thing to consider here actually:
We've written down B2B2... twice.
We need to remove this, so that we're only considering equally likely events.
We now have a list, of 27 equally likely events in our sample space.
Out of these: seven plus seven involve Mr. Jones having a girl.
Therefore we can calculate, the probability, has a girl, is 14 divided by 27,
which is approximately 52%.
And this is closer to our intuition
that, when he has two children, we suspect that it is equally likely,
knowing that one is a boy, that the other could be a girl, is almost 50%.
So now lets relate that to the TED-Ed Frog Riddle.
The video presented, the calculation for variation A.
However, they actually presented the story of variation B
And this can be similar to the Boy or Girl Paradox, knowing that the boy was born on a Tuesday.
So lets go through this calculation:
We know that if we have two frogs,
They can either be: male/male, male/female, female/male, or female/female.
If we know that there is a male frog in the pair,
we remove the female/female pair.
And we end up with a new sample space of three equally likely events:
male/male, male/female, and female/male.
In two of these events, there is a female.
So we can calculate the probability there is a female is equal to two divided by three.
Now this is the probability that is presented in the TED-Ed video.
And it would be true, if you knew that in a pair of frogs there's a male frog, you learn that by random chance.
However, the TED-Ed video presented the variation that you know that there's a male frog, because you heard a croak.
So now we have specify a sample space that involves the male frog's croaking.
We know that there are three different gender pairs that are possible:
male/male, male/female, or female/male.
let's indicate a subscript of one, if we know that the frog croaks, and zero, if the frog does not croak.
We have to consider the sample space of equally likely events.
It could be that the first male frog croaked in a male/male pair.
It could be that the second male frog croaked in the male/male pair
Or, in the male and female pairs, we would have that the male frog is croaking.
The video doesn't tell us actually anything about the probability that male and female frogs are croaking.
So we are going to assume that these are all equally likely events.
The reason we have two different pairs of male/male frogs here,
is that either frog could be croaking.
So in—If we consider this sample space of equally likely events,
the probability that we have a female:
there're two different events out of four.
So that means the probability is equal to two divided by four, which is one half.
So summarizing: the TED-Ed Frog video,
actually presse—they wanted to do a calculation, for variation A.
What's the probability that there's a female, if you know that the pair contains a male frog.
This would be a probability of two thirds.
What they actually presented,
was the variation, where you know there is a male frog, because you heard a croak that is distinctive to male frogs.
And in that case the probability becomes one half.
And intuitively, hearing the male croak changes the sample space.
You are two times as likely to hear a male croak, from a male/male pair,
than from a female/male or a male/female pair.
So I would suggest the TED-Ed Frog Video, as presented, should have a probability of one half.
I understand they wanted to present variation A
Now the key in variation A is that you hear this information by random chance.
Now that's actually pretty hard to do in practice, as you can see that's where they stumbled in the video and made a mistake.
They included the detail: you have to learn somehow in practice.
So ver—Very often in practice, you're most likely going to be presented with variation B.
And that is going to be a probability of one half.
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