Unusual birthday puzzle

[pzhon]

This was posted on usenet a few years ago. Which of the following is more likely?

A) 365 random people have 365 different birthdays.
B) 365^2 random people don't have all 365 birthadys.

The probabilities are close. Ignore February 29th.

[random]

Post deleted by random

[Duke]

--deleted since what was once a question will now look like an unwarranted attack--

[random]

aradsfa;lsk i'm retarded, 6am, going to sleep

[random]

[ QUOTE ]
..will now look like an unwarranted attack

[/ QUOTE ] Certainly not. I said something stupid and it needed to be pointed out. I appreciate you deleting it, though. See ya.

[FrankLu99]

[ QUOTE ]
This was posted on usenet a few years ago. Which of the following is more likely?

A) 365 random people have 365 different birthdays.
B) 365^2 random people don't have all 365 birthadys.

The probabilities are close. Ignore February 29th.

[/ QUOTE ]

ok i am foolish enough to take a shot ( just like how i bluff the river with 8 high)
P(A) == 1/(365!)
P(B) >= 365*((364/365)^(365^2))

Am i close?
I will guess that B is more likely

[bigpooch]

Well, they are close in order of magnitude. If anyone is
so inclined, they can check that the first probability is
roughly bigger by a factor of sqrt(2xpix365). (For those
who are mathematically inclined, it's almost immediate from
Stirling's approximation).

[RiverTheNuts]

im just curious, but how the hell did 2 pi get into a formula with 365 days?? [img]/images/graemlins/smile.gif[/img]

I should really learn this crap

[Duke]

[ QUOTE ]
how the hell did 2 pi get into a formula with 365 days?

[/ QUOTE ]

Pi has nothing to do with the day count. Call it N days and then relate the 2 functions/series/whatever. That relation is where the guy is saying that pi comes from.

I don't know if he's right since I didn't investigate further, but that's what he means. I think.

~D

[bigpooch]

A) The first probability is exactly

P(A) = 365! /( 365^365)

By Stirling's approximation formula, for large n,
n! is roughly sqrt(2 x pi x n) x (n/e)^n where pi is about
3.14159265 and e is about 2.71828183 and so

365! is approximately

sqrt(2 x pi x 365)x(365^365)/(e^365)

and hence,

P(A) is about

sqrt(2 x pi x 365)/(e^365).

By using a better Stirling approximation (asymptotic
expansion of the Gamma function, which is just a function
that extends factorial to all the real numbers), n! is
actually bigger by a factor of

(1 + 1/(12n) + 1/(288 x n^2) - 139/(51840 x n^3) +...)

and so P(A) is really closer to

1.0002283365 x sqrt(2 x pi x 365)/(e^365)

or about

47.9/(e^365)

B) Well, I see where I committed a common error in my
back of the envelope calculation for my previous post!

The second probability p(B) is just

365 x p(everyone is not born on Jan. 1)
+C(365,2) x p(everyone is not born on Jan. 1/2)
+C(365,3) x p(everyone is not born on Jan. 1..3)
+...

It will be clear that the second and subsequent terms are
much smaller than the first term.

The first term is just 365 x (364/365)^(365^2)
=365 x ((364/365)^365)^365.

Now, (364/365)^365 is roughly 1/e; actually, it's about
0.99862857x(1/e). Thus, the first term is close to

365 x (0.99862857/e)^365
or about 221.181/(e^365).

Also, the second term is

(365x364/2) x {((363/365)^365)} ^365

where (363/365)^365 is close to 1/(e^2) and so this term is
in the order of e^-(2x365) which is much smaller than the
first term. One can see that each of the following terms
is then roughly smaller by a factor of e^365 (not quite
because of the combinatorial factor) compared with the
previous term.

In any case, P(B)/P(A) is about 4.61756 and P(B) is really
bigger and these probabilities are much closer than I had
originally thought!


LAZY MAN's APPROXIMATION:

This is the idea that gives one an immediate guess as to
which of these probabilities is bigger. Without caring too
much about how closely 1/e approximates (364/365)^365, one
can see that

P(A) looks like sqrt(2 x pi x 365)/(e^365)

and P(B) is (very!) roughly

365/(e^365).

(I had forgotten the numerator when typing my original
post!). So someone who is really careful and astute can
readily deduce that roughly

P(A)/P(B) = sqrt(2 x pi/365) or about 0.1312. (Actually,
you only need to know that 365 > 2 x pi for guessing that
B is more probable!)

Of course, unfortunately, the 1/e approximation is extremely
crude and hence one has to be careful since raising a crude
approximation to the power of 365 really distorts the
result!