“And even though this problem looks EXACTLY like the other problem minus what seems to
be completely irrelevant information, somehow, we get wildly different results.
How is it that the information that this whale is born on a Tuesday somehow affects whether
the other whale is a male or female?
It makes ZERO intuitive sense.
”
Probability is used everywhere.
You’ve probably seen it in a lot of different contexts, from election polling, to machine
learning, and occasionally in online pop culture.
Say you’re in an Introduction to Probability and Statistics class, and you start learning
about flipping coins and rolling dice and everything seems to have a nice neat formula
that you can fit your problems into, with very simple looking answers.
They ask questions like “What’s the probability of getting 6 heads in a row?”
Well, there’s a 1 in 2 chance of getting heads, 6 heads, one half to the sixth power.
Clean solutions.
Problem is, probability can sometimes be unintuitive.
Probability problems can get tricky.
And answers get very very unintuitive, very quickly, and you get answers that don’t
line up at all with what you might originally think the answer should be.
When I was a student, there was a probability question that always tripped me up.
It really highlights this idea that the answers we get aren’t always the first answers that
come to mind.
I considered it to be the trickiest probability question that I’ve encountered in my introductory
probability class.
But before I bring up this question, I first want to bring up a slightly different problem,
a little bit simpler, but of a similar style:
Suppose you have two whales.
You know that at least one of them is a male whale.
What is the probability that both whales are male?
Take a second to think about this problem intuitively.
If the first thought to come to your mind is 1 in 2, I wouldn’t blame you.
I’d imagine that your thought process went something like this.
You know that one of the whales is male for sure, that’s the given of the problem.
So the probability that they’re both male is just the probability that the other whale
is male too.
That’s just 1 in 2.
But when we actually do this problem out, we actually get a different answer.
Given these two whales, there are four scenarios that can occur.
Both whales could be male, the first one could be male and the second one female, the first
could be female and the second male, or they could both be female.
Note here, that each of these four outcomes are equally likely to occur, each of them
having a 1 in 4 chance of occuring.
Since we’re only looking at the outcomes where at least one of the whales is male,
we don’t have to consider this fourth case, where both whales are female.
For this problem, we are only going to consider these first three cases.
There are three cases of equal probability that we are looking at.
And look, in exactly one of those three, do we have a situation where both whales are
male.
One of three equally likely scenarios, meaning the answer is actually 1 in 3.
Although initially we might think the answer is just one half, the probability that the
other whale is also male, this doesn’t account for the fact that of the four scenarios, there
are two of them that have one of the whales being male and the other being female.
This is because we don’t know whether it’s the first or second whale that is male and
whether it’s the first or second whale that is female, which means we have to account
for this case twice.
This makes it twice as likely that we are in the case where one whale is male and the
other is female.
It’s not that the probability of a whale being born male has changed at all, it’s
just that the distribution of events that we are looking at aren’t necessarily what
we expected.
Yes, there is a 1 in 2 chance that a whale is born male, but laying out all the scenarios,
because we don't know whether the first whale was born male or the second whale was born
male in our given, we have to account for both outcomes, which in the end affects our
final probability calculation.
By the way, if we knew that it was for sure that a specific whale was male, for example
the first whale, then of course we don’t run into this problem.
Now that we know the first whale is male, we only have to consider these two choices.
As you can see, you end up having a 1 in 2 chance that the second whale is also male.
But since we don’t know which of the two whales is the male one, we’re left with
the 1 in 3 answer.
This leads to a bit of a side note, there’s a bit of a language quirk here when you say
the phrase “one of”, as in “one of the whales is male”.
In a natural language like English, if you say something like “one of my kids plays
soccer”, generally speaking you’re referring to a specific kid when you make that statement.
It wouldn’t make much sense for you to mean “one of my kids plays soccer, (oh! but I
don’t remember which one!)” because, why wouldn’t you know which one?
They’re... your kids...
But from an outsider perspective, if some stranger told you, “one of my kids plays
soccer”, then it makes perfect sense for you to be unsure whether it’s their first
kid or their second kid who plays soccer.
I think this ambiguity may feed into some of the confusion in this particular problem.
The wording may be interpreted like it’s from the first scenario, (talking about your
own kids), over the second scenario where you’re just hearing about some stranger’s
kids.
But I digress.
We’ve shown that in this case, the answer is 1 in 3, three choices where there is at
least one male, each with the same probability, and in exactly one of those three situations,
are they both male.
One in Three.
Okay, so now let’s get the moment you’ve been waiting for: The trickiest question I’ve
ever encountered in my introduction class:
Suppose you have two whales.
You know that at least one of them is a male whale who was ALSO BORN ON A TUESDAY, what
is the probability that they are both male whales?
Wait...
Isn’t this just the same problem?
The question only asks for whether or not both whales are male, nothing about the day
of the week the whale was born.
It just doesn’t seem like it’s relevant information that the whale, who we know is
male, was also born on a Tuesday.
There’s no condition that says the other whale must also be born on a Tuesday or can’t
be born on a Tuesday or anything else like that.
The overarching question is the same as the first question: what is the probability that
they are both male whales.
There’s just some extra information about the day of the week that this whale happens
to be born on...
And so we might assume that the answer should be 1 in 3, right?
Just like in the previous question.
But actually... the answer here is... 13 in 27.
What?
Excuse me?
That’s not even close!
How is it that this answer simultaneously failed to be 1 in 3 and also 1 in 2 and also
uses very weird, seemingly unrelated numbers in both the numerator and the denominator?
And even though this problem looks EXACTLY like the other problem minus what seems to
be completely irrelevant information, somehow we get wildly different results.
How is it that the information that this whale is born on a Tuesday somehow affects whether
the other whale is a male or female?
It makes ZERO intuitive sense
But I mean, it checks out.
If we list out all the possible combinations of whales, of being born on any of the seven
days of the week and being born male or female, we find that there are 196 equally likely
outcomes that we are looking at.
Each whale can be either male or female, and be born on one of the 7 days of the week,
giving 14 choices for each whale, and since there are two whales, we square 14 to get
196 different combinations.
Of those 196 different combinations, in 27 of those, we have at least one whale who is
a male born on a Tuesday.
This of course acts as our denominator of the probability.
And of those 27, in 13 of those outcomes, both whales are male.
Here, I’ll break it down even more slowly.
There are 196 equally likely outcomes.
These are evenly distributed amongst the same cases we showed before, Male Male, Male Female,
Female Male, and Female Female.
49 for each case.
Because we know at least one of them is male, just like in the previous question, we can
eliminate this last case right here, leaving us with 147 possible outcomes, again, 49 for
each of the three remaining categories, Male Male, Male Female, and Female Male.
For the latter two cases, where one is a male and the other is a female, we can eliminate
any scenarios where the male is not born on a Tuesday.
So goodbye to any whales born on Sunday, Monday, Wednesday, Thursday, Friday or Saturday.
That leaves us with 1/7th of each of those two cases, 1/7th of 49 is 7 outcomes remaining
for each of the two cases of Male Female and Female Male.
That was fairly easy: We just divided by seven in each of the bottom two cases to get us
7 remaining outcomes each.
Now, let’s pretend for a second that we could apply this same rule to the first case.
Just divide by 7 like the other cases, and get 7 cases.
If we could do this, that would be great!
Now we know we have 21 total possible outcomes that we’re looking at, and 7 of those, the
ones we just counted in this male-male case, have two male whales.
In this case, the answer would just be 7 out of 21, which is, as we would have hoped, 1
in 3.
Alas, it’s not that simple.
See, in the latter two cases, by knowing that the male whale can only be born on a Tuesday,
you actually have quite a bit of information.
You start off with one male and one female.
You know for a fact that the male must be born on Tuesday, and therefore, cannot be
born on any of the other 6 days.
This means you can quite easily eliminate 6/7ths of the outcomes, leaving us with only
7 of the original 49 outcomes each.
But in the first scenario?
You don’t know as much and you can’t make that same assumption.
You can’t simply eliminate the case where the first male whale is born on a Sunday for
example, because technically that’s still possible.
That whale could be born on a Sunday.
You could have the first male whale be born on a Sunday and the second male whale be born
on a Tuesday, and your condition that at least one of the whales was born on a Tuesday is
still met.
You can’t just get rid of outcomes in the same way for the first whale, because the
OTHER one could be the one born on a Tuesday.
Having two male whales makes things much more complicated, almost twice as complicated,
because there are twice as many whales that we can apply our “born on a Tuesday” restriction
to.
In fact, side note: the only reason it’s not 14 which is double 7 and instead is 13,
is because when we’re counting the outcomes, we have to be careful not to double count
the situation where they’re both born on a Tuesday.
It may be tempting just to say, well, if the first whale is born on a Tuesday, then there
are 7 outcomes for the other whale, one for each day of the week, and if the second whale
is the one born on a Tuesday, then there are 7 outcomes for the first whale, 14 total.
But, if you do it like this, there’s the case where both of the male whales are born
on a Tuesday, which you will have counted twice...
So by learning that the male whale was also born on a Tuesday, this scenario where we
have two males suddenly became relatively more likely compared to the other two cases.
Almost twice as many outcomes will fall under this first category as compared with the second
and third categories.
The reason is, because there are so many more cases that a whale is born on a Tuesday in
this first case than in the other two cases, simply because we’re looking at two different
male whales!
That means there’s less certainty in this case, and therefore more possible outcomes
that meet our “two male, where one is born on a tuesday” outcome.
Why is this problem so tricky?
I want to assure you now that your assumptions are 100% correct.
Being born on a Tuesday or any day of the week for that matter does not affect whether
you are male or female, or whether anyone else is male or female or what day of the
week they were born on etc etc.
These things do not affect each other.
But when we talk about independence in probability, we’re not really talking about one thing
affecting another, in this sort of “cause and effect” relationship.
If I drop a vase, it will likely break, cause and effect, one action led to a consequence.
But independence is something different.
Independence is fundamentally about information.
Sure, it’s possible that two things might not affect each other at all whatsoever, but
that doesn’t mean that information about those events can’t give you other pieces
of information about the world.
In this problem, the knowledge that the male whale was also born on a Tuesday, allowed
us to eliminate certain outcomes in some cases much more than in others, changing their probabilities
relative to each other.
That’s one of the things that can make probability unintuitive.
It’s not a model of cause and effect, which is something that’s very intuitive to human
beings.
Instead, it’s a model of how the information that we have about the world can change our
beliefs and understandings about the world in areas that we’re uncertain about.
I’d love for the mathematical world to be a clean place, having complicated questions
with very simple solutions.
There’s a lot of hope that when dealing with probabilities, that any complicated expressions
we have can be simplified, cancelling out the numerator and the denominator for a nice
clean simple answer.
Sometimes that happens, and sometimes you get 13 over 27.
In the real world, we’re usually stuck with using over simplified models to determine
our probabilities, or just outright ignoring most theory and just crunching huge datasets
to give us estimations of our probabilities.
In machine learning and Artificial Intelligence, in many cases we don’t even try.
Likely if you take a statistics or probability course in your Computer Science department,
you’ll likely learn about the Naive Bayes Classifier, which basically simplifies your
world by just assuming everything is independent from each other just so you can have an easier
time computing and calculating your final probabilities.
The problem I showed today can be incredibly frustrating to try and understand, and even
more frustrating to get wrong on an assignment or an exam.
But I share this problem because I think it demonstrates a very important idea, and that
idea is that Probability isn’t always intuitive.
By that I don’t mean it’s impossible to understand, as much as I mean that the first
answer that comes to your mind may not always be the correct one.
There are many pitfalls and errors you can make in probability problems, including things
like, simplifying the question too much, overcounting, or simply not anticipating some of the ways
that new information can affect your knowledge.
Even with what seems to be a fairly simple problem can turn out to have very non-trivial
answers.
In these cases, it’s important to remember fundamentals, like using Bayes Rule and explicitly
drawing out your scenarios using charts or venn diagrams, and remembering not to rush
to oversimplify problems.
I wanted to share this problem because I’ve thought about this problem a lot.
I remember knowing what the answer was because I plugged the numbers into the right equation,
but as much as I tried, I couldn’t explain to myself or any of my classmates why we got
such an un-intuitive answer.
Eventually, we just typed up our answer and handed it in without really understanding
it.
That’s the thing: Our classes and our exams were usually set up in a way that incentivized
us to rush through problems without fully understanding them, plugging things blindly
into formulas without realizing why we got the answers we did.
That’s why I wanted to create this channel: I want to give students the resources to learn
and crucially, to understand.
Being able to leave each video not just to go out and do things yourself, but understand
it to a point where you can go and explain it to others.
If you think that’s a worthy goal, feel free to subscribe and we’ll get more content
for you in the future!
For now, thanks for watching!
Hope to see you again sometime!