---
title: "R Problem due Tuesday 9/18"
author: "YOUR NAME"
output: pdf_document
---

Acknowledgements: ADD NAMES HERE IF ANYONE ASSISTED YOU.  

**Email knitted pdf file to tleise@amherst.edu by 4pm on the due date.**

Read Problem 2.28. Then answer the problem by completing the parts below. Here is the code provided in the problem to simulate birthdays from a group of 23 people, outputting a frequency table of how many people share each birthday (with days of the year numbered 1 to 365). 

```{r}
bday.table <- table(sample(1:365,23,replace=TRUE))
bday.table
```

We want to know whether or not at least two people shared a birthday in the simulated data, so we ask whether any numbers greater than 1 occur in the table. The code suggested in the problem statement only checks whether exactly 2 share a birthday. We'd like to allow 2 or more to share a birthday, so we'll use "any" rather than "%in%" for this problem.
```{r}
any(bday.table>1)
```

a. Estimate the probability that at least two people have the same birthday in a room of 23 people through 10000 simulations like the one above. Feel free to check your answer by calculating the theoretical formula in R.

```{r}
set.seed(6)

```

> ANSWER

b. Use simulations like what you did in part a to estimate the number of people needed so that the probability of a match is greater than 95%. You should check using multiple seeds, as the estimate will vary a bit from simulation to simulation, or else verify your answer by computing the formula.

```{r}


```

> ANSWER

c. Use similar simulations to estimate the probability that at least three people share the same birthday in a room of 50 people.

```{r}


```

> ANSWER

d. Estimate the number of people needed so that the probability of having three people or more with the same birthday is at least 95%. Check your answer using multiple runs of the simulation, as the estimated probability will vary a bit.

```{r}


```

> ANSWER