Exercises.R

#Exercises

1.1
cyan <- 3
magenta <- 5
yellow <- 7

# Assign a variable `p` as the probability of choosing a cyan ball from the box
p <- cyan / (cyan + magenta + yellow)

# Print the variable `p` to the console
p


1.2
# `p` is defined as the probability of choosing a cyan ball from a box containing: 3 cyan balls, 5 magenta balls, and 7 yellow balls.
# Using variable `p`, calculate the probability of choosing any ball that is not cyan from the box
1-p


1.3
cyan <- 3
magenta <- 5
yellow <- 7

# The variable `p_1` is the probability of choosing a cyan ball from the box on the first draw.
p_1 <- cyan / (cyan + magenta + yellow)

# Assign a variable `p_2` as the probability of not choosing a cyan ball on the second draw without replacement.
p_2 <- (magenta + yellow) / (cyan-1 + magenta + yellow)

# Calculate the probability that the first draw is cyan and the second draw is not cyan.
p_1 * p_2


1.4
cyan <- 3
magenta <- 5
yellow <- 7

# The variable 'p_1' is the probability of choosing a cyan ball from the box on the first draw.
p_1 <- cyan / (cyan + magenta + yellow)

# Assign a variable 'p_2' as the probability of not choosing a cyan ball on the second draw with replacement.
p_2 <- (magenta + yellow) / (cyan + magenta + yellow)

# Calculate the probability that the first draw is cyan and the second draw is not cyan.
p_1 * p_2





#Exercise 2.2
cyan <- 3
magenta <- 5
yellow <- 7

# Assign the variable 'p_yellow' as the probability that a yellow ball is drawn from the box.
p_yellow <- yellow / (cyan + magenta + yellow)

# Using the variable 'p_yellow', calculate the probability of drawing a yellow ball on the sixth draw. Print this value to the console.
p_yellow


#Exercise 2.3
# Assign the variable 'p_no6' as the probability of not seeing a 6 on a single roll.
p_no6 <- 5/6

# Calculate the probability of not seeing a 6 on six rolls.
p_no6^6


#Exercise 2.4
# Assign the variable `p_cavs_win4` as the probability that the Cavs will win the first four games of the series.
p_cavs_win4 <- (6/10)^4

# Using the variable `p_cavs_win4`, calculate the probability that the Celtics (the other team) win at least one game in the first four games of the series.
1 - p_cavs_win4


#Exercise 2.5
# This line of sample code simulates four random games where the Celtics either lose or win. Each game is independent of other games.
simulated_games <- sample(c("lose","win"), 4, replace = TRUE, prob = c(0.6, 0.4))

# The variable 'B' specifies the number of times we want the simulation to run. Let's run the Monte Carlo simulation 10,000 times.
B <- 10000

# Use the `set.seed` function to make sure your answer matches the expected result after random sampling.
set.seed(1)

# Create an object called `celtic_wins` that first replicates the sample code generating the variable called `simulated_games` for `B` iterations and then tallies the number of simulated series that contain at least one win for the Celtics.
celtic_wins <- replicate(B, {
  simulated_games <- sample(c("lose","win"), 4, replace = TRUE, prob = c(0.6, 0.4))
  any(simulated_games %in% "win")
})

# Calculate the frequency out of B iterations that the Celtics won at least one game. Print your answer to the console.
mean(celtic_wins)




#Exercise 3.1
# Assign a variable 'n' as the number of remaining games.
n <- 6

# Assign a variable 'l' to a list of possible game outcomes, where 0 indicates a loss and 1 indicates a win for the Cavs. 
l <- list(c(0,1))

# Create a data frame named 'possibilities' that contains all possible outcomes for the remaining games.
possibilities <- expand.grid(rep(sample(l), n))
possibilities

# Create a vector named 'results' that indicates whether each row in the data frame 'possibilities' contains enough wins for the Cavs to win the series.
results <- rowSums(possibilities) >= 4
results

# Calculate the proportion of 'results' in which the Cavs win the series. Print the outcome to the console.
mean(results)


#Exercise 3.2
# The variable `B` specifies the number of times we want the simulation to run. Let's run the Monte Carlo simulation 10,000 times.
B <- 10000

# Use the `set.seed` function to make sure your answer matches the expected result after random sampling.
set.seed(1)

# Create an object called `results` that replicates the sample code for `B` iterations and tallies the number of simulated series that contain at least four wins for the Cavs.
results <- replicate(B, {
  simulated_games <- sample(c(0, 1), 6, replace = TRUE)
  sum(simulated_games) >= 4
})


# Calculate the frequency out of `B` iterations that the Cavs won at least four games in the remainder of the series. Print your answer to the console.
mean(results)


#Exercise 3.3
# Let's assign the variable 'p' as the vector of probabilities that team A will win.
p <- seq(0.5, 0.95, 0.025)

# Given a value 'p', the probability of winning the series for the underdog team B can be computed with the following function based on a Monte Carlo simulation:
prob_win <- function(p){
  B <- 10000
  result <- replicate(B, {
    b_win <- sample(c(1,0), 7, replace = TRUE, prob = c(1-p, p))
    sum(b_win)>=4
  })
  mean(result)
}

# Apply the 'prob_win' function across the vector of probabilities that team A will win to determine the probability that team B will win. Call this object 'Pr'.
Pr <- sapply(p, prob_win)

# Plot the probability 'p' on the x-axis and 'Pr' on the y-axis.
plot(p, Pr)


#Exercise 3.4
# Given a value 'p', the probability of winning the series for the underdog team B can be computed with the following function based on a Monte Carlo simulation:
prob_win <- function(N, p=0.75){
  B <- 10000
  result <- replicate(B, {
    b_win <- sample(c(1,0), N, replace = TRUE, prob = c(1-p, p))
    sum(b_win)>=(N+1)/2
  })
  mean(result)
}

# Assign the variable 'N' as the vector of series lengths. Use only odd numbers ranging from 1 to 25 games.
N <- seq(1, 25, 2)

# Apply the 'prob_win' function across the vector of series lengths to determine the probability that team B will win. Call this object `Pr`.
Pr <- sapply(N, prob_win)

# Plot the number of games in the series 'N' on the x-axis and 'Pr' on the y-axis.
# Let's assign the variable 'p' as the vector of probabilities that team A will win.
p <- seq(0.5, 0.95, 0.025)

# Given a value 'p', the probability of winning the series for the underdog team B can be computed with the following function based on a Monte Carlo simulation:
prob_win <- function(p){
  B <- 10000
  result <- replicate(B, {
    b_win <- sample(c(1,0), 7, replace = TRUE, prob = c(1-p, p))
    sum(b_win)>=4
  })
  mean(result)
}

# Apply the 'prob_win' function across the vector of probabilities that team A will win to determine the probability that team B will win. Call this object 'Pr'.
Pr <- sapply(p, prob_win)

# Plot the probability 'p' on the x-axis and 'Pr' on the y-axis.
plot(p, Pr)



#Exercise 4.6
# Assign a variable 'female_avg' as the average female height.
male_avg <- 69

# Assign a variable 'female_sd' as the standard deviation for female heights.
male_sd <- 3

# Determine the height of a man in the 99th percentile of the distribution.
qnorm(0.99, male_avg, male_sd)




#Exercise 5.2
# Use the `set.seed` function to make sure your answer matches the expected result after random sampling.
set.seed(1)

# The variables 'green', 'black', and 'red' contain the number of pockets for each color
green <- 2
black <- 18
red <- 18

# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- green / (green+black+red)

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- (red+black) / (green+black+red)

#Create a model to predict the random variable `X`, your winnings from betting on green.
X <- sample(c(17, -1), 1, replace = TRUE, prob = c(p_green, p_not_green))

# Print the value of `X` to the console
X



#Exercise 5.3
# The variables 'green', 'black', and 'red' contain the number of pockets for each color
green <- 2
black <- 18
red <- 18

# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- green / (green+black+red)

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1-p_green

# Calculate the expected outcome if you win $17 if the ball lands on green and you lose $1 if the ball doesn't land on green
#Note to self: Expected Value = ap + b(1-p)
17 * p_green + -1 * p_not_green



#Exercise 5.4
# The variables 'green', 'black', and 'red' contain the number of pockets for each color
green <- 2
black <- 18
red <- 18

# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- green / (green+black+red)

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1-p_green

# Compute the standard error of the random variable
# Note to self: SE = sqrt(no of draws) * std deviation of the numbers in the 'urn'
# Formula: |b-a| * sqrt(p * (1-p))
abs(-1 - 17) * sqrt(p_green * p_not_green)



#Exercise 5.5
# The variables 'green', 'black', and 'red' contain the number of pockets for each color
green <- 2
black <- 18
red <- 18

# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- green / (green+black+red)

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1-p_green

# Use the `set.seed` function to make sure your answer matches the expected result after random sampling
set.seed(1)

# Define the number of bets using the variable 'n'
n <- 1000

# Create a vector called 'X' that contains the outcomes of 1000 samples
X <- sample(c(17, -1), n, replace = TRUE, prob = c(p_green, p_not_green))

# Assign the sum of all 1000 outcomes to the variable 'S'
S <- sum(X)

# Print the value of 'S' to the console
S


#Exercise 5.6
# The variables 'green', 'black', and 'red' contain the number of pockets for each color
green <- 2
black <- 18
red <- 18

# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- green / (green+black+red)

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1-p_green

# Define the number of bets using the variable 'n'
n <- 1000

# Calculate the expected outcome of 1,000 spins if you win $17 when the ball lands on green and you lose $1 when the ball doesn't land on green
n * (17 * p_green + -1 * p_not_green)


#Exercise 5.7
# The variables 'green', 'black', and 'red' contain the number of pockets for each color
green <- 2
black <- 18
red <- 18

# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- green / (green+black+red)

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1-p_green

# Define the number of bets using the variable 'n'
n <- 1000

# Compute the standard error of the sum of 1,000 outcomes
sqrt(n) * (abs(-1 - 17) * sqrt(p_green * p_not_green))




#Exercise 6.1
# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- 2 / 38

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1-p_green

# Define the number of bets using the variable 'n'
n <- 100

# Calculate 'avg', the expected outcome of 100 spins if you win $17 when the ball lands on green and you lose $1 when the ball doesn't land on green
avg <- n * (17*p_green + -1*p_not_green)

# Compute 'se', the standard error of the sum of 100 outcomes
se <- sqrt(n) * (17 - -1)*sqrt(p_green*p_not_green)

# Using the expected value 'avg' and standard error 'se', compute the probability that you win money betting on green 100 times.
1- pnorm(0, avg, se)



#Exercise 6.2
# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- 2 / 38

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1-p_green

# Define the number of bets using the variable 'n'
n <- 100

# The variable `B` specifies the number of times we want the simulation to run. Let's run the Monte Carlo simulation 10,000 times.
B <- 10000

# Use the `set.seed` function to make sure your answer matches the expected result after random sampling.
set.seed(1)

# Create an object called `S` that replicates the sample code for `B` iterations and sums the outcomes.
S <- replicate(B, {
  X <- sample(c(17, -1), n, replace = TRUE, prob = c(p_green, p_not_green))
  sum(X)
})

# Compute the average value for 'S'
mean(S)

# Calculate the standard deviation of 'S'
sd(S)



#Exercise 6.3
# Calculate the proportion of outcomes in the vector `S` that exceed $0
mean(S > 0)



#Exercise 6.5
# Use the `set.seed` function to make sure your answer matches the expected result after random sampling.
set.seed(1)

# Define the number of bets using the variable 'n'
n <- 10000

# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- 2 / 38

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1 - p_green

# Create a vector called `X` that contains the outcomes of `n` bets
X <- sample(c(17, -1), n, replace = TRUE, prob = c(p_green, p_not_green))

# Define a variable `Y` that contains the mean outcome per bet. Print this mean to the console.
Y <- mean(X)
Y



#Exercise 6.6
# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- 2 / 38

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1 - p_green

# Calculate the expected outcome of `Y`, the mean outcome per bet in 10,000 bets
#The expected outcome of a single spin is the sum of the individual outcomes for winning or losing multiplied by their chances of occurring. 
(17 * p_green) + (-1 * p_not_green)


#Exercise 6.7
# Define the number of bets using the variable 'n'
n <- 10000

# Assign a variable `p_green` as the probability of the ball landing in a green pocket
p_green <- 2 / 38

# Assign a variable `p_not_green` as the probability of the ball not landing in a green pocket
p_not_green <- 1 - p_green

# Compute the standard error of 'Y', the mean outcome per bet from 10,000 bets.
abs(-1 - 17) * sqrt(p_green * p_not_green)/sqrt(n)



#Exercise 6.8
# We defined the average using the following code
avg <- 17*p_green + -1*p_not_green

# We defined standard error using this equation
se <- 1/sqrt(n) * (17 - -1)*sqrt(p_green*p_not_green)

# Given this average and standard error, determine the probability of winning more than $0. Print the result to the console.
1 - pnorm(0, avg, se)



#Exercise 6.9
# The variable `n` specifies the number of independent bets on green
n <- 10000

# The variable `B` specifies the number of times we want the simulation to run
B <- 10000

# Use the `set.seed` function to make sure your answer matches the expected result after random number generation
set.seed(1)

# Generate a vector `S` that contains the the average outcomes of 10,000 bets modeled 10,000 times
S <- replicate(B, {
  X <- sample(c(17, -1), n, replace = TRUE, prob = c(2/38, 36/38))
  mean(X)
})

# Compute the average of `S`
mean(S)

# Compute the standard deviation of `S`
sd(S)