This course will provide you with an intuitive and practical introduction into Probability Theory. You will be able to learn how to apply Probability Theory in different scenarios and you will earn a "toolbox" of methods to deal with uncertainty in your daily life. The course is split in 5 modules. In each module you will first have an easy introduction into the topic, which will serve as a basis to further develop your knowledge about the topic and acquire the "tools" to deal with uncertainty. Additionally, you will have the opportunity to complete 5 exercise sessions to reflect about the content learned in each module and start applying your earned knowledge right away. The topics covered are: "Probability", "Conditional Probability", "Applications", "Random Variables", and "Normal Distribution". You will see how the modules are taught in a lively way, focusing on having an entertaining and useful learning experience! We are looking forward to see you online!
The Birthday Problem
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Del curso dictado por University of Zurich
An Intuitive Introduction to Probability
468 calificaciones
De la lección
Application
We will discuss some fascinating every-day applications of probability. In addition to entertaining examples, we will also review very serious applications from finance and law.
Conoce a los instructores
Karl SchmeddersProfessor of Quantitative Business Administration
Department of Business Administration
Welcome to our next module in an Intuitive Introduction to Probability
decision-making in an uncertain world.
In this module I want to show you a bunch of applications of probabilities.
As I said at the very beginning of this course,
many people tell me "I hate probability."
And I think that attitude comes from that many people don't really see
cool, real-world applications that are really relevant
in everyday decision-making, or relevant
to the well-being of our societies.
And here I really want to convince you that there are cool applications.
We want to start with some playful applications.
Today in the first lecture we're going to look at the Birthday Problem.
And then later on we get into some more serious stuff
from business or the law.
So, let's just dive into the Birthday Problem.
What is that about?
You go to a party and there's a bunch of people in the room.
And here is now the question: what is the probability
that there's at least one pair of people at this party
who have birthday on the same day of the year?
Two people having birthday on January 18th
or March 22nd or July 1st.
And then the related question: How many people
do you have to have at this party, so that this probability
of at least one pair of birthday people in the room
is larger than a half, larger than 50%?
These two questions together give us a Birthday Problem.
Before I tell you how to think about this
here's an in-class quiz question for you.
Let's go back to the second question: How many people do you need
to have in a room so that the probability of at least one pair
is larger than 50%?
Here I have a bunch of options of possible answers.
Exactly one of them is correct.
Please have a go at it, and then afterwards I will tell you
how we can calculate these probabilities.
Welcome back from the in-class quiz question.
Now I want to show you how we can calculate these probabilities.
I've prepared an Excel sheet called Birthday Problems.
Now, before I show you the calculations in that sheet,
I want to tell you a little bit about the strategy
on how we calculate the probabilities that I mentioned before.
It's actually a really tricky question to think about
how many people may have birthday on the same day
because maybe there's one pair, maybe there are two pairs
maybe there are 3 or more pairs, maybe there are triplets.
There are so many possibilities that it quickly
gets totally overwhelming.
And I can tell you, I have no idea on how to calculate
those probabilities that way.
But I can show you much more elegant, much more simpler way.
Let's use the Complement Rule, remember this was one
of the key rules that we learned about in the every first module.
So the probability of the complement of an event
is 1 minus the probability of the event.
What's the opposite of having at least one pair,
or 2 pairs or some more?
Ha! The opposite is very easy, very well defined.
All of us at this party, all the people in this room
having birthday on different days.
And it turns out, that probability can be calculated quite easily.
So now, come with me to the Excel sheet and I'll show you
how we can calculate that probability.
Here we are now at the spreadsheet for the Birthday Problem
that I prepared
Before we get into the calculation, I need to state my assumptions
which are listed over here in column E.
First, we assume there's no February 29th so we assume
there are only 365 days in the given year.
All of those days are equally likely and we assume
all the people in the room are independent.
That means your birthday doesn't affect mine
our birthdays don't affect anyone else's and so on.
And as I now said before, we are not yet calculating the probability
of the event we really care about, that there's at least one pair,
instead we're looking at the complement.
What's the probability that all of us in the room
have their birthday on different days?
We look at Person 1, Person 1 can choose
his or her birthday any day in the year.
It get's interesting with Person 2.
This person now has only 364 out of 365 days left, why?
The first person picked the day, Oh, that's my birthday
and now we want the second person to be different
well, that leaves only 364.
And so now I start calculating these probabilities here.
One, anything goes for the first person times 364 divided by 365.
So this number here, the .99726 is the probability that a pair
has birthday on different days.
Now the third person enters the party.
What's the probability that her birthday is different
than of the first 2 people?
Ah! Two days are taken in the year. That means 363 days are left.
So we multiply with 363 divided by 365 and get this probability
a little less that 99.2%.
You now see as more and more people enter the room
the probability of being different gets a little smaller, little smaller
and independence, we multiply all of them.
And so now what happens is that this product
gets smaller and smaller and smaller.
Now the amazing thing here is that the probability
drops below a half already at 23.
So, as soon as there are 23 people in the room
and assuming our assumptions apply, then the probability of all of us
having birthday on a different day is already below a half.
That now means of 23 people, the probability of at least one pair
is larger than a half.
that's very counter-intuitive and quite surprising.
So often people tell me this can't be true, how can this be?
Now, I calculated probabilities for many other numbers
including 45, 45 is under 6%
And now for 45, I also prepared a Monte Carlo simulation.
How does this work now? Let's go here to the second sheet.
Here now I randomly pick 45 numbers and that's very easy in Excel.
There's this cool function RANDBETWEEN and then I do 1 SIM
I call on 365 on this Swiss computer, in the United States it's 1 comma 365
whatever it is in your language, and this gives me
a random number between 1 and 365.
I do this for all 45 people.
And then I compare, look at all pairs.
So here I compare Person 2 to Person 1.
I compare Person 3 to Person 1 and 2.
Here I compare Person 10 to the first 9.
If I see a match between Person 8 and 5 as I indicate here
then I get a 1, otherwise the spreadsheet shows me a zero.
Now at the top here, I count the number of matches we have in our spreadsheet.
And now I can recalculate the sheet, either by clicking a button here
or in a Windows computer, easily F9 and I create 45 new birthdays
a new party, 45 new people, and I check is there a pair?
Look at this, right now I have 2 pairs
and so I guess there's at least 1 match.
I do it again, 3 pairs. I do it again, another 2 pairs.
Two pairs, 4 pairs, wow! Look at this, 3 pairs, 1 pair.
Now, you see, it's hard to hit a zero.
Do it for a while, and eventually you will also occasionally
hit a zero, I can tell you in less than 6% of all times.
Before I return the spreadsheet, I want to do one last detour.
People find this so strange, how can it be that with 23 people
we already have a likelihood larger than a half, of having a pair?
Now let me go the third and last sheet on my Excel spreadsheet here.
Now we change the problem. Here the question is now:
What is the probability that at least 2 people
have a birthday today, or the complement
that nobody of us has a birthday today?
This is a fundamentally different problem. Why?
I fix the day. I said let's look at birthdays today.
This now rules out you and me having birthdays
on the same day of the year that's different than today.
That's in the previous version of the problem that was a hit
that was a match, that was a pair.
And there are so many days, 365, where you and I
could have a birthday together.
Here now, that birthday has to be today.
Now the probability calculation changes.
Here now, notice the different definition.
Now we are looking at the probability that all of us
do not have a birthday today.
Now, for each of us, the probability of not having a birthday today
is 364 divided by 365, it doesn't decrease.
You and I can have both birthday yesterday.
That's still not a match here, so now the probability
as you see here in my spreadsheet
is 364 divided by 365 to the power of the number of people in the room.
Independence, your probability not today
364 divided by 365 times my birthday, not being today
364 times 365 times and so on for all of them.
And now you may think, oh I need 183 people for this to be 50/50.
Why do people think this? Oh, that's
365 divided by a half is 182.5, so if a 183 people
this will be about a half. Wrong.
Because from those 183 people, quite a few
will have birthdays on the same days.
And therefore, it's not that we have covered half a year
for sure with 183 people, actually less.
You can now play with this, and the surprising result is that
you need 253 people at your party to have a probability
of more than a half, that at least one of them has a birthday today.
That's a very different calculation.
So let me sum this up one more time.
In this last instance of the problem, the question was
what's the probability that you have someone having
a birthday today, so today and a person's birthday matches.
That's very different than in the original Birthday Problem
at the beginning here, where it can be any day of the year.
And so I think this helps a lot of people to get over
this counter-intuitive result that you only need 23 people
to have a match on some day of the year.
Not today, but some day of the year.
This wraps up the Excel spreadsheet discussion.
Play around with it a little bit more if you want.
And now let's summarize everything in these slides and return there.
Here we are back from our Excel sheet.
Let me briefly summarize our calculations.
We made the following assumptions: there are 365 days in a year.
So we are ignoring February 29th.
We assume that all days are equally likely
and finally, that we are all independent of each other.
Your birthday doesn't affect mine, nor anyone else's in the room.
And then we just saw the probability now of the complement
of the original event, namely all of us having birthday on different days
is one big long multiplication.
The first person can have birthday on any given day.
The second person now needs to be different
as the first person has taken a day, for example January 1st.
So the second person has 364 out of 365 days left.
So I multiply 1 times 364 divided by 365.
Now the second person comes along
in addition to the first, so in total the third person.
Now this person cannot have birthday on the first 2 days
that Person 1 and Person 2 had a birthday on.
So that's 363 out of 365 which we multiply
with the previous product and so on.
And now you go on until however many people are in this room.
To summarize, here are the results.
On the one hand with 10 people, there's more than 88% chance
that all of us that are in the room, if there are only 10 of us
have birthday on a different day.
With 23 people, that already drops below a half.
Put differently, for the original event that we talked about
what's the probability of having at least one pair?
With only 23 people, that probability already exceeds a half.
So the answer to that quiz question was 23,
the smallest of the choices given.
Now, notice once you have 60 or more people in the room,
the probability of all of them having different birthdays
drops below 1%.
So, that means next time you are at at a party
you may want to offer that bet.
Yeah, I think you will be able to surprise some people.
And that already brings us to the takeaways.
In addition to this rather surprising,
perhaps counter-intuitive result of the Birthday Problem
we also have seen an application of the Complement Rule.
The Complement Rule sometimes can be very, very helpful
when the calculation of the probability of an event
is very, very tedious, or we think maybe
the Complement Rule allows me easier calculation.
That's the first application in this module.
Come back for more fun in the next few lectures.
Thank you.
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