### Introduction to Simulating Coin Tossing

Simulation provides a straightforward way of approximating probabilities. For simulating a Game of chance of coin tossing, one simulates a particular random experiment (coin tossing, dice roll, and/or card drawing) a large number of times. The probability of an outcome is approximated by the relative frequency of the outcome in the repeated experiments.

## Table of Contents

The use of simulation experiments to better understand probability patterns is called the **Monte Carlo Method**.

### Practical Example: Simulating Coin Tossing Experiment

Let person “A” and person “B” play a simple game involving repeated tosses of a fair coin. In a given toss, if the head is observed, “A” wins $1 from “B”; otherwise if the tail is tossed, “A” gives $1 to “B”. If “A” starts with zero dollars, we are interested in his fortune as the game is played for 50 tosses.

#### Simulating The Game in R using sample() Function

For the above scenario, one can simulate this game using the R function “sample()”. A’s winning on a particular toss will be $1$ or $-1$ with equal probability. His winning on 50 repeated tosses can be considered to be a sample of size 50 selected with replacement from the set {$1, -$1}.

option(width=60) sample(c(-1, 1), size = 50, relapce = T) # output [1] -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 [26] 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 1 1

#### Graphical Representation of the Simulations

One can graphically represent the outcome, as coded below. Note that the results will be different for each compilation of the code as samples are drawn randomly.

results <- sample( c(-1, 1), size = 50,replace = TRUE) x = table(results) names(x) = c("loss", "win") barplot(x)

### Extended Example

Suppose “A” is interested in his cumulative winnings as he plays this game. One needs to score his toss winnings in the variable $win$. The function “`cumsum()`

” will compute the cumulative winnings of the individual values and the cumulative values are stored in “`cam.win`

“.

win = sample(c(-1, 1), size = 50, replace = T) cum.win = cumsum(win) # Output for different execution [1] -1 -2 -1 -2 -1 0 1 0 -1 0 1 0 1 2 1 2 1 2 3 4 5 6 7 6 5 [26] 4 5 4 5 6 7 6 7 8 9 8 7 6 5 4 5 4 3 2 1 0 -1 -2 -3 -4 [1] 1 2 1 0 1 2 1 2 1 2 1 2 3 4 5 4 5 6 7 8 7 8 7 6 7 6 5 4 5 4 3 4 3 4 5 4 3 2 [39] 3 4 3 4 3 2 3 4 5 6 5 4

Extending and plotting the sequence of cumulative winnings for 4 games. For four games, the win/loss score is plotted in four combined (2 by 2) graphs, to visualize the situation in all four games.

par(mfrow = c(2, 2))

for (j in 1:4){

win = sample( c(-1, 1), size = 50, replace = TRUE)

plot(cumsum(win), type = “l”, ylim = c(-15, 15))

abline(h=0)

}

The horizontal line in each graph is drawn at break-even. The points above the horizontal line show the win situation while the information below the horizontal line shows the loss to the player.

One can make a customized function for the situation discussed above. For example,

# customized function winloss <- function (n=50){ win = sample(c(-1,1), size = n, replace = T) sum(win) } # Insights about win/ loss situation F = replicate(1000, winloss() ) table(F) # output F -22 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 3 6 7 22 28 52 60 78 82 128 93 103 95 72 50 48 34 24 8 3 20 22 24 1 2 1 par(mfrow = c(1, 1) ) plot(table(F))