Introduction
I’m finishing up the fourth week of my master’s in statistics program at Oklahoma State University. I’ve already learned so much about hypothesis testing, SAS (yuck), probability, and R.
My probability course is very theoretical, but often we refer to real world examples using dice, decks of cards, and other games.
I’m constantly looking for ways to expand my knowledge of R, so I decided to take ideas surrounding probability and dice and illustrate them in R with a blog post.
So without further ado, let’s load up the trusty tidyverse and get rolling!
Libraries
library('tidyverse')
Rolling a Single Die
Starting off in the simplest way - I want to simulate the roll of a die using R. The base function sample was built for such a task. sample generates a random number or series of numbers within a bound you set, replacing the numbers at your discretion. If you want each roll of the die to be independent, simply set the replace argument to true.
sample(1:6,1)
## [1] 4
Looks like we rolled a 4! Notice how you don’t need to use the replace argument if your only sampling a single number with each use of the function.
Relative Frequency with a Die
Relative frequency is the fraction of outcomes in which an event occurs. The relative frequency might be equal to the probability of an event, but it might not.
You would expect a die to roll each of the six possible numbers 1/6 of the time. That’s the probability of each event in the sample space.
The relative frequency is the number of times a die rolls a certain number, or event.
Let’s see the relative frequency of 50 independent rolls of a single die and see how close they are to the expected probability of 1/6.
# use the sample function to simulate 50 independent rolls of a die
single_die_1 <- sample(1:6, 50, replace=T)
# order the values into a dataframe named sd_1_freq
sd_1_freq <- data_frame(num = 1:6, roll = table(single_die_1)/50)
# plot the frequency data
ggplot(sd_1_freq) +
geom_bar(aes(num,roll),
stat = "identity",
color = "darkslateblue",
fill = "darkslateblue") +
geom_hline(yintercept = (1/6),
col = "firebrick",
size = 1
) +
scale_x_continuous(name = "Result",
breaks = sd_1_freq$num,
labels = sd_1_freq$num) +
ylab("Relative Frequency") +
ggtitle("50 Roll Relative Frequency") +
theme_minimal()
The red bar denotes the expected 1/6 or 0.166 probability of each event. Some of our relative frequency results are close to their expected probability, but most are not.
Relative frequency approaches the probability as the sample size increases.
To illustrate that this is true, let’s increase the sample size from 50 rolls to 50,000 rolls.
# use the sample function to simulate 50,000 independent rolls of a die
single_die_2 <- sample(1:6, 50000, replace=T)
# order the values into a dataframe named sd_1_freq
sd_2_freq <- data_frame(num = 1:6, roll = table(single_die_2)/50000)
# plot the frequency data
ggplot(sd_2_freq) +
geom_bar(aes(num,roll),
stat="identity",
color = "darkslateblue",
fill = "darkslateblue"
) +
geom_hline(
yintercept = (1/6),
col = "firebrick",
size = 1
) +
scale_x_continuous(name = "Result",
breaks = sd_2_freq$num,
labels = sd_2_freq$num
) +
ylab("Relative Frequency") +
ggtitle("50,000 Roll Relative Frequency") +
theme_minimal()
After 50,000 rolls, the relative frequency is almost exactly the expected probability. We could continue move the relative frequency closer and closer to the probability value by increasing the sample size even more. But I’ll call this one
Summing Two Dice
Have you ever played the casino game Craps? In Craps, you roll two dice and sum them to determine if you’ve automatically won (7 or 11), lost (2, 3, or 12) or if you need to roll again.
So let’s start by rolling two dice.
sum(sample(1:6,1), sample(1:6,1))
## [1] 6
That was easy! Now let’s do it 10,000 times and plot the relative frequency of each result! Because our sample size will be so high (10,000), the relative frequencies should be negligibly close to the probabilities of each outcome.
# create empty dataframe of 10,000 values
roll_vector <- vector("double", 10000)
# for each element in roll_vector
for (i in seq_along(roll_vector)) {
# the element equals the sum of two dice
roll_vector[i] = sum(sample(1:6,1), sample(1:6,1))
}
# turn the resulting vector into a dataframe to plot
roll_df <- as_data_frame(table(roll_vector)/10000)
# name the variables roll, and freq
colnames(roll_df) <- c("roll", "freq")
# turn the table-created roll variable into a numeric
roll_df$roll <- as.numeric(roll_df$roll)
# plot the relative frequencies
ggplot(roll_df) +
geom_bar(aes(roll, freq),
stat = "identity",
color = "mediumpurple4",
fill = "mediumpurple4"
) +
scale_x_continuous(name = "Sum of Dice",
breaks = roll_df$roll,
labels = roll_df$roll
) +
ylab("Relative Frequency") +
ggtitle("10,000 Two-Dice Roll Relative Frequency") +
theme_minimal()
Generalizing with a Function
So we’ve recorded the relative frequency of one and two-die rolls, can we do it for a 100 die roll? How about a 10,000 Die roll? and What if I want to roll the die 67 times instead of 50, or 670,000 times instead of 50,000 times?
To efficiently create plots for any input, we need to create a function. I’ll call the function roll_em!
# Create the function "roll_em" to sum any number of dice for any number of rolls and produce a relative frequency plot as the output.
# naming the function
roll_em <- function(n_dice, n_roll) {
# check for input errors, both inputs must be non-zero integers
if (n_dice == 0 | is.na(n_dice) | n_dice %% 1 != 0) {
stop("Enter a non-zero integer for the number of dice to roll")
}
else if (n_roll == 0 | is.na(n_roll) | n_roll %% 1 != 0) {
stop("Enter a non-zero integer for the number of rolls to perform")
}
else {
# if no errors are detected, move on to the body of the function
# declare a counter, i, and set it to zero
i <- 0
# create roll vector of user-specified length
roll_vector <- vector("double", n_roll)
# while the counter doesn't equal the user-specified length of roll_vector, loop through the following commands
while (i <= length(roll_vector)) {
# create a new dice_vector of user-specified length
dice_vector <- vector("double", n_dice)
# for each element j in dice vector, roll a single die and record the value
for (j in seq_along(dice_vector)) {
dice_vector[j] = sample(1:6,1)
}
# sum the values in the dice vector and put that value into the current roll_vector element
roll_vector[i] <- sum(dice_vector)
# add one iteration to the roll_vector counter
i <- i + 1
# repeat the sequence
}
# once all elements of the roll_vector are filled, create a roll dataframe
roll_df <- as_data_frame(table(roll_vector)/n_roll)
# name the columns, or variables, roll and freq (frequency)
colnames(roll_df) <- c("roll", "freq")
# ensure the roll numbers are numerics, instead of characters (default in table)
roll_df$roll <- as.numeric(roll_df$roll)
# plot the output
ggplot(roll_df) +
geom_bar(aes(roll, freq),
stat="identity",
color = "midnightblue",
fill = "midnightblue"
) +
scale_x_continuous(name = "Sum of Dice",
breaks = roll_df$roll,
labels = roll_df$roll
) +
ylab("Relative Frequency") +
ggtitle("Roll 'EM! Results") +
theme_minimal()
# conclude else statement function
}
# conclude roll_em function
}
Now that we’ve made roll_em, let’s check to see if it works!
Testing Roll ‘EM!
A generic entry:
roll_em(5,200)
An entry designed to give an error:
roll_em(0,1)
## Error in roll_em(0, 1): Enter a non-zero integer for the number of dice to roll
Another error entry:
roll_em(2,500.1)
## Error in roll_em(2, 500.1): Enter a non-zero integer for the number of rolls to perform
An obscure entry:
roll_em(502,7)
A large number of rolls:
roll_em(5, 91572)
Looks like roll_em works as we expect it to! It’s tempting to put enormous inputs, 1 billion rolls of 100 billion dice, but you’ll probably crash R doing it.
What a fun little project to practice making functions and better visualize relative frequencies.
I’m starting to work through Hands-On Programming with R in my free time and it looks like there are a few more casino examples coming my way. So, look forward to a blog post about playing cards and one about slot machines in the near future!
Until next time,
- Fisher
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