Assignment4RMarkdown.Rmd

---
title: "Assignment 4"
author: "Prithvi Abhishetty"
date: "2022-11-18"
output: html_document # you can change to other output format if you want
---

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

# 1. Probability theory
....

# 2. Finite probability spaces

## 2.1 (Q1)

$\binom{22}{z} \left(\frac{3}{10} \right)^z \cdot \left(\frac{7}{10}\right)^{22-z}$

## 2.1 (Q2)

```{r}
num_red_balls <- 3
num_blue_balls <- 7
total_draws <- 22
prob_red_spheres <- function(z){
  total_balls <- num_red_balls + num_blue_balls
  prob <- choose(total_draws,z) * (num_red_balls/total_balls)^z * (num_blue_balls/total_balls)^{total_draws-z}
  return(prob)
}

prob_red_spheres(10)
```
## 2.1 (Q3)

```{r}
num_reds=seq(22)
prob_by_num_reds <- data.frame(num_reds, prob=prob_red_spheres(num_reds))

prob_by_num_reds %>% head(3)
```

## 2.1 (Q4)

```{r}
ggplot(prob_by_num_reds, aes(x=num_reds, y=prob)) + geom_line() + xlab('Number of reds') + ylab('Probability') + theme_bw()
```

## 2.1 (Q5)

```{r}
sample(10, 22, replace=TRUE)

## Setting the random seed just once
set.seed(0)
for(i in 1:5){
print(sample(100,5,replace=FALSE))
# The result may well differ every time
}
## Resetting the random seed every time
for(i in 1:5){
set.seed(1)
print(sample(100,5,replace=FALSE))
# The result should not change
}
```
```{r}
num_trials<-1000 # set the number of trials
set.seed(0) # set the random seed
sampling_with_replacement_simulation<-data.frame(trial=1:num_trials) %>%
mutate(sample_balls = map(.x=trial, ~sample(10,22, replace = TRUE)))
# generate collection of num_trials simulations

sampling_with_replacement_simulation <- sampling_with_replacement_simulation %>%
mutate(num_reds = map_dbl( .x=sample_balls, ~ sum(.x<=3) ) )
```


## 2.1 (Q6)

```{r}
num_reds_in_simulation<-sampling_with_replacement_simulation %>%
pull(num_reds)

# we extract a vector corresponding to the number of reds in each trial
prob_by_num_reds<-prob_by_num_reds %>%
mutate(predicted_prob=map_dbl(.x=num_reds,~sum(num_reds_in_simulation==.x))/num_trials)
# add a column which gives the number of trials with a given number of reds
```
## 2.1 (Q7)

```{r}
prob_by_num_reds %>%
rename(TheoreticalProbability=prob, EstimatedProbability=predicted_prob) %>%
pivot_longer(cols=c("EstimatedProbability","TheoreticalProbability"),
names_to="Type",values_to="count") %>%
ggplot(aes(num_reds,count)) +
geom_line(aes(linetype=Type, color=Type)) + geom_point(aes(color=Type)) +
scale_linetype_manual(values = c("solid", "dashed"))+
theme_bw() + xlab("Number of reds") + ylab("Probabilities")
```


## 2.2 (Q1)

```{r}
# Step 1
set.seed(0) # set the random seed
# Step 2
num_trials <- 1000 # set the number of trials
sample_size <- 10
# Step 3
sampling_without_replacement_simulation <- data.frame(trial=1:num_trials) %>%
mutate(sample_balls=map(.x=trial,~sample(100,sample_size,replace = FALSE)))
# generate collection of num_trials simulations

# Step 4
sampling_without_replacement_simulation <- sampling_without_replacement_simulation %>%
mutate(num_reds=map_dbl(.x=sample_balls, ~sum((.x<=50))),
num_blues=map_dbl(.x=sample_balls, ~sum((.x>50)*(.x<=80))),
num_greens=map_dbl(.x=sample_balls, ~sum((.x>80))),
)

# Step 5
sampling_without_replacement_simulation <- sampling_without_replacement_simulation %>%
mutate(mun_minimum=pmin(num_reds, num_blues, num_greens))
# Step 6
proportion_zero = mean(sampling_without_replacement_simulation$mun_minimum==0)
print(proportion_zero)

```
## 2.2 (Q2)

```{r}
num1 <- choose(50,10) + choose(70,10) + choose(80,10) -
choose(20,10) - choose(30,10) - choose(50,10)
num2 <- choose(100,10)
probs_A <- num1/num2
print(probs_A)
```