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.