The use of simulation experiments to better probability patterns is called the Monte Carlo method. This is an example.

#### Tossing Coin Game

Tom and Jerry play a simple game of tossing a coin. If given heads, Tom pays Jerry a dollar. Otherwise, Jerry pays Tom a dollar.

``````set.seed(1212)

# how many tosses?
number_of_tosses = 50

# simulate n tosses
# 1 = head, -1 = tail
(tosses <- sample(c(1, -1), size = number_of_tosses, replace = TRUE, prob = c(.5, .5)))``````
``````##   -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  1  1  1  1 -1 -1 -1  1 -1  1  1  1  1 -1 -1  1  1 -1 -1  1
##  -1  1  1 -1``````

Let’s observe some games’ progression, from the perspective of Tom.

``````library(purrr)

# convert game to a function
new_game <- function(n = 1) {
tosses = sample(c(1, -1),
size = n,
replace = TRUE,
prob = c(.5, .5))
}

# progress of 4 different games
par(mfrow = c(2, 2))
walk(1:4, ~ {
plot(cumsum(new_game(number_of_tosses)), type = "l", ylab = "Winnings")
abline(h = 0, col = "red", lty = 3)
})`````` Let’s calculate the winnings distribution.

``````library(dplyr)
library(ggplot2)
library(ggthemes)

old <- theme_set(theme_tufte() + theme(text = element_text(family = "Menlo")))

# modify function above
new_game <- function(n = 1) {
tosses = sample(c(1, -1),
size = n,
replace = TRUE,
prob = c(.5, .5))
# return winnings
sum(tosses)
}
# repeat many times
reps <- 10e3 %>% rerun(new_game(number_of_tosses)) %>% unlist()

# bind into data frame
reps %>%
as_data_frame() %>%
ggplot(aes(value)) +
geom_histogram(binwidth = 1) +
scale_x_discrete(limits = seq(-24, 24, 4))`````` Question: Why there isn’t any odd number?

We can also calculate the chances where Tom break even, either from simulation result above, or using binomial distribution. They are pretty close.

``````# how many breakeven
length(which(reps == 0)) / length(reps)``````
``##  0.1163``
``````# approx. form using binomial distribution
dbinom(25, size = 50, prob = 0.5) ``````
``##  0.1122752``

What is the likely number of tosses Peter will be winning in a game of 50 tosses? What is the likely maximum fortune Peter will be winning in a game of 50 tosses?

``````library(tidyr)

# modify function above
new_game <- function(n = 1) {
tosses = sample(c(1, -1),
size = n,
replace = TRUE,
prob = c(.5, .5))
# track progress
cum.win = cumsum(tosses)
# return
list(final = sum(tosses),
max = max(cum.win))
}
new_game(number_of_tosses)``````
``````## \$final
##  12
##
##  50
##
## \$max
##  13``````
``````# repeat
set.seed(1234)
reps <- 10e3 %>% rerun(new_game(number_of_tosses))

reps %>%
bind_rows() %>%
# collapse columns into one
gather(var, val) %>%
ggplot(aes(val)) +
geom_histogram(binwidth = 1) +
facet_wrap(~ var, ncol = 1,
labeller = labeller(
var = c(final = "Final Winnings", 