# Birthday Paradox

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• March 16th 2010, 09:51 PM
jrkevinfang1234
Birthday Paradox
Apparently, there are two methods of solving this probability problem. The first, using p(n) = 1 - (365^n falling / 365^n) makes sense to me; however, there is an alternate solution using the Taylor series of e^x.

Here's the full solution on Wikipedia:
Birthday problem - Wikipedia, the free encyclopedia

I got lost at the transition from
http://upload.wikimedia.org/math/7/4...c4ca4959f8.png

to

http://upload.wikimedia.org/math/d/9...c059edb90e.png

What's happening here?
• March 16th 2010, 10:15 PM
Random Variable
let $x=-x$

then $e^{-x} \approx 1 -x$

so $1 - \frac{1}{365} \approx e^{-1/365}$

$1-\frac{2}{365} \approx e^{-2/365}$

and so on
• March 16th 2010, 10:16 PM
Drexel28
Quote:

Originally Posted by jrkevinfang1234
Apparently, there are two methods of solving this probability problem. The first, using p(n) = 1 - (365^n falling / 365^n) makes sense to me; however, there is an alternate solution using the Taylor series of e^x.

Here's the full solution on Wikipedia:
Birthday problem - Wikipedia, the free encyclopedia

I got lost at the transition from
http://upload.wikimedia.org/math/7/4...c4ca4959f8.png

to

http://upload.wikimedia.org/math/d/9...c059edb90e.png

What's happening here?

$e^x\approx 1+x\implies e^-x\approx 1-x$ and so $\left(1-x\right)(1-y)(1-z)\cdots(1-t)\approx e^{-x}\cdot e^{-y}\cdot e^{-z}\cdots e^{-t}$
• March 26th 2010, 05:29 PM
stringtheery
Interesting birthday paradox.
I have an interesting situation. My friend and I have the same birthday. One of her siblings and one of my siblings have the same birthday. She has 2 sibs and I have 6. What are the odds and probability of that occuring? Thanks for responding.