Simulation

In this notebook, we will be simulating coin tosses by using for loop in R and will be showing the results by using a line graph. Through this exercise, we hope that you understand the law of large numbers.

There is a very useful function in R called sample() which randomly selects a value inside a vector. The code works like:

sample(vec, size = , replace = TRUE)

vec is the vector that contains all the values that you would like to use in the simulation, size specifies the number of simulations, and replace specifies whether you are going to do the simulation with replacement or without replacement. We usually do simulation wit replacement, so set this to be TRUE.

For example:

dice <- c('One', 'Two','Three','Four','Five','Six')
sample(dice, size=1, replace = TRUE)

## [1] "Five"

If you run the code for a multiple number of times, you realize that the outcome of the code changes everytime you run the code. To make the problem simpler, we will let:

Head = 1
Tail = 0

and do the simulation.

# This will be the coin that we will use
coin <- c(0, 1)

# Specify two empty vectors at first to store the number of toss and the outcome
times <- c()
prob <- c()

for (n in 1:100) {
  # n will be the number of toss
  times <- append(times, n)
  # Simulate n number of coin tosses
  out <- sample(coin, size = n, replace = TRUE)
  # probability of getting a head
  p <- sum(out) / n
  prob <- append(prob, p)
}

The list of probabilities are in prob and the number of tosses are in times, so we would now create a line chart that shows the result:

plot(times, prob, type = 'l', col = 'blue', main = 'Coin toss simulation',
     xlab = 'Number of coin toss', ylab = 'Probability of getting a head')
abline(h=0.5, lty=2, col='red')

As you can see from the diagram, the probability of getting a head gets really close to 0.5 as we increase the number of coin tosses.