Homework_3_Wildfires_Rosskopf_Golombek.Rmd

---
title: "Homework 3 - Wildfire"
author: "Martina Rosskopf & Nina Golombek"
date: "25 November 2018"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```
## Excersize:

Assume that lightning strike causes wildfires with a probability of 1/1000. 
You observe a wildfire and an eyewitness claims that it was lit by lightning. 
What is the probability that this is correct, knowing that eyewitness reports are 
reliable 80% of the time?

### Given Values:

$P(L)$ = P(Wildfire caused by lightning) = 1/1000 = 0.0001

$P(L_c)$ = P(Wildfire caused by something else) = 1 - P(L) = 1 - 0.0001 = 0.999


$P(E|L)$ = P(Eyewitness is correct|Wildfire caused by lightning) = 80% = 0.8

$P(E|L_c)$ = P(Eyewitness is correct|Wildfire caused by something else)   1 - P(E|L) = 1 - 0.8 = 0.2

### Wanted:

$P(L|E)$ = P(Wildfire caused by lightning|Eyewitness is correct)

### Formula: Bayes Law

The first step of finding the solution for this exercise was to look at the formular of Baye's:
$$P(A|B) = \frac{P(B|A) * P(A)}{P(B)}$$
$$P(B) = P(B|A) * P(A) + P(B|A_N) * P(B_N)$$

## Result:
By looking at the formular and rereading the given values in the questioning, we came to the conclusion that $P(A|B)$ is the probability for a wildfire caused by lightning on condition that the eyewitness is right. Which is the probablity we are looking for. Rewriting the formular with different variables.

$$P(L|E) = \frac{P(E|L)*P(L)}{P(E)} = \frac{0.8 * 0.001}{0.2006} = 0,00398 = 0.398 \% $$
$$P(E) = P(E|L) * P(L) + P(E|L_c) * P(L_c) =  0.8 * 0.001 + 0.2 * 0.999 = 0.2006 $$

Now we can write the formular in an R code to calculate the final probability.
First we need to define the known probabilities:

```{r}
p_l = 1 / 1000         #P(L)   
p_lc = 1 - p_l         #P(L_c)
p_e_l = 0.8            #P(E|L)
p_e_lc = 1 - p_e_l     #P(E|L_c)
```
Next we get the solution for the exercise with upper formular.
```{r}
p_l_e = p_e_l*p_l / (p_e_l*p_l + p_e_lc*p_lc)
print(p_l_e)
```
We then can see that the probability for a wildfire caused by lightning, given that the eyewitness is right, is (rounded) $P(L|E) = 0.004$ or 0.4%.




  


```{r}



```