# Toss a fair coin 10 times

coinToss <- sample(c("H", "T"), 10, replace = TRUE)
coinToss

# Estimate the probability that you observe Heads for the 
# third time on the 10th toss.

sum(coinToss == "H")

sum(coinToss == "H") == 3

sum(coinToss[1:9] == "H") == 2

sum(coinToss == "H") == 3 & sum(coinToss[1:9] == "H") == 2

event <- replicate(10000, {
  coinToss <- sample(c("H", "T"), 10, replace = TRUE)
  (sum(coinToss == "H") == 3 & coinToss[10] == "H")
})

mean(event)

# Roll a pair of dice 1000 times.
# What is the probability that the value of the second die
# is greater than the value of the first die?

help(sample)

n <- 1000; n
x <- sample(1:6, size = n, replace = TRUE)
head(x)

y <- sample(1:6, size = n, replace = TRUE)
head(y)

z <- (x < y)
head(z)

mean(z) # answer 15/36

15/36

# Now replicate the simulation of the 1000 rolls, 1000 times.

B <- 1000; B  # number of replications

n <- 1000; n  # number of rolls of the pair of dice

z <- replicate(B, {
x <- sample(1:6, size = n, replace = TRUE)
y <- sample(1:6, size = n, replace = TRUE)
z <- (x < y)
mean(z)
})

head(z)

hist(z)