r-function |
Description |
|---|---|
length |
Compute the length of a vector |
mean |
Compute the mean |
sum |
Compute the sum |
cat |
Concatenate strings and variable values for formatted print |
sample(x, size, replace=FALSE), prob=NULL) |
Randomly sample x uniformly with the number of samples = size. If prob is provided, length(prob)=length(x), and prob should sum to 1. |
table |
Tabulate the frequency counts of distinct values. |
barplot(x) |
Plot barplot of the 1-D data x: arguments: main: main title; xlab: x-label; ylab: y-label; col: color |
factorial(n) |
n! |
permutations(n,m), combinations(n,m) |
List all permutations (combinations) of n objects taken m at a time (n>m). Accepts an optional argument v for indicating the set of permuted n elements. |
nrow |
Find the number of rows in a matrix, dataframe or other rectangular data structure |
In this code:
We calculate the probability of drawing a Heart from a deck of four suits (sample space).
We simulate random events such as a coin toss and rolling a six-sided die.
We simulate multiple die rolls and visualize the resulting probability distribution.
We calculate the probability of a specific outcome (such as, rolling a 3).
# Set a seed for reproducibility
set.seed(42)
# Define a sample space (e.g., a deck of cards)
sample_space <- c("Hearts", "Diamonds", "Clubs", "Spades")
# Calculate the probability of drawing a Heart from the sample space
probability_heart <- sum(sample_space == "Hearts") / length(sample_space)
cat("Probability of drawing a Heart:", probability_heart, "\n")Probability of drawing a Heart: 0.25 # Simulate a random event (e.g., coin toss)
coin_toss <- sample(c("Heads", "Tails"), size = 1)
cat("Result of a random coin toss:", coin_toss, "\n")Result of a random coin toss: Heads # Simulate rolling a six-sided die
die_roll <- sample(1:6, size = 1)
cat("Result of rolling a die:", die_roll, "\n")Result of rolling a die: 5 # Simulate multiple die rolls and visualize the probability distribution
num_rolls <- 1000
die_rolls <- sample(1:6, size = num_rolls, replace = TRUE)
# Calculate the relative frequencies for each outcome
relative_frequencies <- table(die_rolls) / num_rolls
relative_frequenciesdie_rolls
1 2 3 4 5 6
0.171 0.192 0.164 0.157 0.154 0.162 # Calculate the probability of rolling a 3
probability_roll_3 <- relative_frequencies[3]
cat("Probability of rolling a 3:", probability_roll_3, "\n")Probability of rolling a 3: 0.164 # Visualize the probability distribution with a bar plot
barplot(relative_frequencies, main = "Probability Distribution of Die Rolls",
xlab = "Die Face", ylab = "Probability", col = "lightblue")
R provides a built-in function to calculate factorial. You can use the factorial() function in R to compute the factorial of a number.
n <- 5
factorial_result <- factorial(n)
cat("Factorial of", n, "is", factorial_result, "\n")Factorial of 5 is 120 Replace the value of n with the number for which you want to calculate the factorial, and the factorial() function will return the result.
To do this, we can use the permutations function from the gtools package. For any list of size n, this function computes all the different permutations P(n,r) we can get when we select r items. Here are all the ways we can choose two numbers from a list consisting of 1,2,3:
library(gtools)
permutations(3, 2) [,1] [,2]
[1,] 1 2
[2,] 1 3
[3,] 2 1
[4,] 2 3
[5,] 3 1
[6,] 3 2Notice that the order matters here: 3,1 is different than 1,3. Also, note that (1,1), (2,2), and (3,3) do not appear because once we pick a number, it can’t appear again.
To get the actual number of permutations, one can use the R-function nrow() to find the total number of rows in the output of permutations:
library(gtools)
nrow(permutations(3,2))[1] 6Alternatively, we can add a vector v to indicate the objects that a permutation is performed on. If you want to see five random seven digit phone numbers out of all possible phone numbers (without repeats), you can type:
all_phone_numbers <- permutations(10, 7, v = 0:9) # Use digits 0, 1, ..., 9
n <- nrow(all_phone_numbers)
cat("total number of phone numbers n = ", n, "\n")total number of phone numbers n = 604800 # Randomly sample 5 phone numbers
index <- sample(n, 5)
all_phone_numbers[index,] [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 8 9 5 1 6 0 3
[2,] 4 0 2 3 5 7 8
[3,] 0 4 6 1 3 2 7
[4,] 5 8 2 6 4 0 3
[5,] 7 5 1 0 9 2 6The code all_phone_numbers[index,] extract the matrix all_phone_numbers with rows indexed by index, and all columns because no specific column index is given after ,.
Instead of using the numbers 1 through 10, the default, it uses what we provided through v: the digits 0 through 9.
How about if the order doesn’t matter? For example, in Blackjack if you get an Ace and a face card in the first draw, it is called a Natural 21 and you win automatically. If we wanted to compute the probability of this happening, we would enumerate the combinations, not the permutations, since the order of drawn cards does not matter.
combinations(3,2) [,1] [,2]
[1,] 1 2
[2,] 1 3
[3,] 2 3In the second line, the outcome does not include (2,1) because (1,2) already was enumerated. The same applies to (3,1) and (3,2).
To get the actual number of combinations, one can do
nrow(combinations(3,2))[1] 3(optional) Of course, one can define a R-function to calculate a permutation number.
# Function to calculate permutation (nPr)
nPr <- function(n, r) {
if (n < r) {
return(0)
} else {
return(factorial(n) / factorial(n - r))
}
}
nPr(3,2)[1] 6# Function to calculate combination (nCr)
nCr <- function(n, r) {
if (n < r) {
return(0)
} else {
return(factorial(n) / (factorial(r) * factorial(n - r)))
}
}
nCr(3,2)[1] 3