Question

Please help me write these in R script / Code

1, Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car; behind the others, goats. You pick a door, say #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He then says to you, "Do you want to pick door #2?" What is the probability of winning the car if you use the strategy of first picking a random door and then switching doors every time? Note that the host will always open a door you did not pick, and it always reveals a goat.

2, A famous coin tossing game has the following rules: The player tosses a coin repeatedly until a tail appears or tosses it a maximum of 1000 times if no tail appears. The initial stake starts at 2 dollars and is doubled every time heads appears. The first time tails appears, the game ends and the player wins whatever is in the pot. Thus the player wins 2 dollars if tails appears on the first toss, 4 dollars if heads appears on the first toss and tails on the second, 8 dollars if heads appears on the first two tosses and tails on the third, and so on. Mathematically, the player wins 2k dollars, where k equals the number of tosses until the first tail. What is the probability of profit if it costs 15 dollars to participate?

3, In a random thirteen-card hand from a standard deck, what is the probability that none of the cards is an ace and none is a hear

4, In six coin tosses, what is the probability of having a different side come up with each throw, that is, that you never get two tails or two heads in a row

5, A random five-card poker hand is dealt from a standard deck. What is the chance of a flush (all cards are the same suit)?

Answer 1

Answer 1: R script will be :

# Load the package ggplot2, if it is not already loaded
library( ggplot2 )
library( dplyr )

Game_show <- function(n=1000, switch=TRUE) {

# Initialize the win vector, where we'll store wins
win <- rep(NA, n)

# Define the possible prizes and 'bad prizes' resulting in a loss
prizes <- c("Car", "Goat1", "Goat2")
badprizes <- c("Goat1", "Goat2")

for (i in 1:n) {
  
# Player picks a door randomly
x <- sample(1:3, 1)
# Randomly assign each one of the prizes to a 'door'
doorsetup <- sample(prizes,3)
if(switch) {
if(doorsetup[x]=="Car") {
reveal <- sample(badprizes,1)   
revealpos <- which(doorsetup==reveal)   
} else if (doorsetup[x]=="Goat1") {
revealpos <- which(doorsetup=="Goat2")
} else if (doorsetup[x]=="Goat2") {
revealpos <- which(doorsetup=="Goat1") }
y <- doorsetup[-c(x, revealpos)]
} else if (!switch) {
y <- doorsetup[x]
}

# Determine whether player won based on the value of y
if (y=="Car") {
win[i] <- 1
} else {win[i] <- 0}
}

n_wins <- sum( win )
n_losses <- n - sum( win )

# Print & graph the results of simulation
data <- data.frame(result = c("loss","win"),
count = c( n_losses, n_wins ),
relativefrequency = c( n_losses/n, n_wins/n ),
stringsAsFactors = FALSE )
plot <- ggplot(data, aes(x=result, y=relativefrequency)) +
geom_bar(stat = "identity")
  
list( data = data,
plot = plot )
}

Using the Game_show function to simulate playing the Game_show game 100 times

### Using the function

set.seed(123)

# Strategy: You always stick with your original door choice

Game_show(n=100, switch=FALSE)

# Strategy: You always switch doors when given the chance to switch

Game_show(n=100, switch=TRUE)

Answer 2: R script will be: