counting problems

[serene bridge]

I am doing a problem where I want to find the smallest number of people n such that the proability of two people having the same birthday is > 1/2. I am doing this by calculating P(2 people (of n total people) have same birthday) = 1 - P(no 2 people have the same birthday)

Then P(no 2 people have the same birthday) = #(ways to have no people (n people total) have same birthday)/#(ways to assign birthdays without restrictions) = {365 \choose n}/#(weak permutations of n into 365 parts). However, this is not correct for some reason as the denominator should be 365^n...

so where did I go wrong?

[sharp dawn]

{365 \choose n}/#(weak permutations of n into 365 parts)

this is strange to me

I also don't know what weak permutations of n into r parts means

but you can do this without fancy formulas. Just think of choosing a day that hasn't been chosen for each successive person.

P(person 1 does not have a bday already chosen) = 365/365

For person 2, it's 364/365.

you can generate the formula from that

[proper anvil]

You said without restrictions meaning that people can have the same birthday in the denom so each person has 365 choices

Also each person is distinct which is why u have to use permutation

[serene bridge]

right i get that

but why doesnt the weak compositions method work

is it cuz the numerator doesnt treat the people as distinct

whereas the denominator does?

[proper anvil]

Yes both do

Both consider them distinct

[serene bridge]

i dont think the top does

its simply choosing n days of 365 days

not assigning the n days to each person

[proper anvil]

But each person is distinct

[serene bridge]

i mean that doesnt really matter for the question

[proper anvil]

Giving Tommy first of Jan is different than giving Janet first of Jan

So it is distinct

[serene bridge]

lmao

sure

but the numerator isnt doing tjat

[proper anvil]

It is u are finding the amount of ways to give a day to each person

Bc the days are distinct and so are the people

[serene bridge]

what?

365 choose n

chooses n distinct days

but it doesnt assign a day to a person

it just says that these are the days these people have birthdays

not that person A has a particular birthday

[plush girder]

I can show you how to solve it if you can remind me of the notation using the previous answer to show something

I think it’s called sequential

Anyway I have a equation its just not in terms other people would know

[proper anvil]

Yes but since you ordered the denominator you have to order the numerator

Yes your way works but then we would have to divide the bottom by n! Since we aren’t ordering it

So in conclusion you can do choosing but then u would also have to not order the bottom

Your method works it’s just u multiply both the top and bottom by 1/n!

In order to get rid of ordering

By ordering I mean u consider each person distinct

Hope this helps

😁

[plush girder]

I think the term I am looking for is nested functions

Idk what the thing I made before was called but this is what it is in y=f(x)

So this could be correct but the answer using it is 252

[serene bridge]

Oh yeah that’s what I’m asking

But I’m not sure that the denominator is ordered

Actually I’m not convinced that the denominator is ordered either

Cuz in stars and bars the boxes (days in this case) are distinct

Whereas the balls (people) are identical

But the numerator also distinguishes between different days of the year

[serene bridge]

In reference to here ^

[plush girder]

Here is a alternate form I made

However this alternative form won’t be equal to the correct answer since it’s assuming you can use parts of a person

It does if you round though

@serene bridge where did you learn to separate it into P and 1-P

Wait I just realized I misunderstood the question

So my formulas only work if you are calculating for 1 specific birthday

[serene bridge]

wdym

like the complement of an event?

[proper anvil]

Yes but with the 365^n denominator it also orders the people(people are distinct) howerver with your method u consider people non distinct. That means that you still consider dates distinct. So to calculate numerator you first find number of ways you can choose days for n people. That would result in 365^n, however this value orders the dates (e.g the first day chosen and the second day chosen is different form choosing the second day first & the first day different). You would only order the dates if the people are distinct. So in order to get rid of ordering you would need to divide the numerator by n! the amount of ways to order the denominator(dates).

[serene bridge]

wait no the numerator only counts the number of ways to have people have distinct birthdays

oh so ur saying it should be P(no 2 people have the same birthday) = #(ways to have no people (n people total) have same birthday)/#(ways to assign birthdays without restrictions) = {365 \choose n}/(#(weak compositions of n into 365 parts) * n!)?

[proper anvil]

well you have to multiply the numerator by n! as well to cancel out

but yes

i think you did it correctly

but u looked at an unsimplified solution

in which they considered people distinct

but you didn't

[proper anvil]

answer*

[serene bridge]

no the answer is $\frac{{365 \choose n}}{365^n}$

[turbid basin]

Henry

[serene bridge]

my answer is just wrong

[proper anvil]

Do u have sol

Hmm

Where did u get answer from

[sharp dawn]

I explained the answer up here

[serene bridge]

Yes

I understand the solution

My question is why my solution is wrong

But I asked SE and figured it out

It’s cuz the events don’t have equal probability, and counting number of occurrences of an event divided size of sample space only works if the events all have equal probability