Thread: Birthday Paradox

1. 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

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2. let $\displaystyle x=-x$

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

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

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

and so on

3. 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

to

What's happening here?
$\displaystyle e^x\approx 1+x\implies e^-x\approx 1-x$ and so $\displaystyle \left(1-x\right)(1-y)(1-z)\cdots(1-t)\approx e^{-x}\cdot e^{-y}\cdot e^{-z}\cdots e^{-t}$

4. 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.