Wildfire Lightning_JMK II.Rmd

---
title: "Wildfire Lightning"
author: "Jan Kärstens"
Date: "22.11.2018"
output: html_document
---

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



In our case scenario we observe a wildfire. The possibility for it to be caused by a lightning is 1/1000. An eyewitness tells us that he even observed the lightning, causing the fire. This statement has a probability of 0.8 to be true.

Giving us the probabilities:
```{r Probabilities}
#Probabilities
P_w <- 0.8 #Probability of eyewitness TRUE
P_wc <- 1-P_w #Probability of eyewitness FALSE; c-counter
P_f <- 0.001 #Probability of Wildfire caused by Lightning
P_fc <- 1-P_f #Probability of Wildfire caused by everything Else; c-counter
```

Knowing this, we ask ourselves what's the probability for this event of a lighnting caused fire and the eyewitness telling the truth?:

## P(w | f)

Resolving Bayes Law we get the formula:

##P(w | f) = (P(f | w) / (P(f | w) + P(f | wc))

In the numerator we have to consider both cases: 
Eyewitness is right and the fire was caused by a lightning (P(f | w)) and the case he is wrong and the fire was caused by something else (P(fc | wc)).

w - eyewitness TRUE;
wc - eyewitness FALSE ("c-counter");
f - fire caused by lightning;
fc - fire NOT caused by lightning ("c-counter");

```{r P_w_f}
#Calculation of P(w | f)
P_w_f <- ((P_f * P_w)) / (((P_f * P_w)) + ((P_fc * P_wc))) #P(w | f)
returnValue(P_w_f) #Giving Value calculated
```
On the other hand, we can also calculate the counter-event probability. The case that the witness is false and the fire was caused by something else (P(fc | wc)):

```{r P_w_f}
#Calculation of P(w | fc)
P_w_fc <- ((P_fc * P_wc)) / (((P_fc * P_wc)) + ((P_f * P_w)))
returnValue(P_w_fc)
```
To ensure we calculated correctly, we add up the both probabilities:
```{r Sum}
#Checking Probabilities' sum equals 1
sum <-  P_w_f + P_w_fc
returnValue(sum)
```