Untitled.Rmd

---
title: "The Birthday Problem"
author: "Fernando Hoces de la Guardia"
date: "9/28/2018"
output:
  html_document: default
  pdf_document: default
  word_document: default
---

```{r}
n.pers = 100
set.seed(1234)
```


# The birthday problem

The probability that at least two people in this room have the same birthday feels like somthing like `r round(n.pers/365, 2)`

Is is $\frac{1}{365} \times `r n.pers` = `r 1/365 * n.pers`$

## Analytical solution    


\begin{align}   
 1 - \bar p(n) &= 1 \times \left(1-\frac{1}{365}\right) \times \left(1-\frac{2}{365}\right) \times \cdots \times \left(1-\frac{n-1}{365}\right) \nonumber  \\  &= \frac{ 365 \times 364 \times \cdots \times (365-n+1) }{ 365^n } \nonumber \\ &= \frac{ 365! }{ 365^n (365-n)!} = \frac{n!\cdot\binom{365}{n}}{365^n}\\
p(n= `r n.pers`) &= `r  round(1 - factorial(n.pers) * choose(365,n.pers)/ 365^n.pers, 3)`  \nonumber
\end{align}