How to Apply Bernoulli Sampling in R (Example)

A Bernoulli sampling refers to a process of selecting outcomes based on a binary random variable, typically modeled as either a success (1) or a failure (0), with a specified probability of success. It is widely used in various fields such as statistics, machine learning, and experimental design.

In statistics, Bernoulli sampling is often used to model binary outcomes, such as success or failure, in experiments. In machine learning, it can be applied to simulate probabilistic events. In biology, it is useful in modeling genetics and survival studies, where only two outcomes are possible.

Using R, we can simulate Bernoulli trials, visualize their outcomes, and analyze their statistical properties like the probability of success and the distribution of outcomes. These simulations help us understand random processes and their applications in real-world scenarios.

The article will cover the following topics:

Let’s begin our exploration of Bernoulli sampling in R!

Install and Load Required Package

To visualize our Bernoulli sampling results, we’ll use the ggplot2 library. This library provides powerful tools for creating clear and professional data visualizations in R.

# Install ggplot2 package
install.packages("ggplot2")
 
# Load ggplot2 library
library(ggplot2)

Understanding Bernoulli Sampling

A Bernoulli trial is a random experiment with exactly two possible outcomes: “success” (1) or “failure” (0). The probability of success is denoted by p, and the probability of failure is 1 – p. Each trial is independent of the others.

For example, imagine flipping a biased coin where the probability of heads (success) is 0.7, and the probability of tails (failure) is 0.3. Each flip represents a Bernoulli trial.

In R, we can simulate Bernoulli sampling using the rbinom() function. This function generates random samples from a binomial distribution, where we set the number of trials to 1 for each Bernoulli trial.

Simulating Bernoulli Sampling in R

Let’s start by simulating Bernoulli sampling with a probability of success p = 0.7 (i.e., a biased coin with a 70% chance of heads).

# Define the parameters
n_trials <- 1000      # Number of trials
p_success <- 0.7      # Probability of success
 
# Simulate Bernoulli trials
bernoulli_samples <- rbinom(n_trials, size = 1, prob = p_success)
 
# Print the first few samples
head(bernoulli_samples)
# [1] 1 0 1 0 1 1

The rbinom() function generates random samples from a binomial distribution. The size = 1 argument ensures each trial has only one outcome, and the prob = p_success argument sets the probability of success (heads).

Visualizing the Results

We can now visualize the results of the Bernoulli sampling to understand the distribution of successes and failures. A simple bar plot can show the relative frequency of success (1) and failure (0).

# Convert samples to a data frame
samples_data <- data.frame(Sample = bernoulli_samples)
 
# Plot the distribution of Bernoulli outcomes
ggplot(samples_data, aes(x = factor(Sample))) +
  geom_bar(fill = "#1b98e0") +
  labs(title = "Bernoulli Sampling Distribution", x = "Outcome", y = "Count") +
  theme_minimal()

The bar plot shows the distribution of successes (1) and failures (0) after 1000 trials. We can clearly see the expected result: most of the outcomes are successes (1), as the probability of success was set to 0.7.

Simulating Multiple Bernoulli Trials

To understand variability in Bernoulli trials, we can simulate multiple sets of Bernoulli trials and compare their outcomes. Each set of trials will have the same probability of success, but the results will differ due to randomness.

# Define the number of trials per set
n_sets <- 5
 
# Simulate multiple sets of Bernoulli trials
bernoulli_sets <- replicate(n_sets, rbinom(n_trials, size = 1, prob = p_success))
 
# Convert data to long format for ggplot2
sets_data <- data.frame(Set = rep(1:n_sets, each = n_trials),
                        Sample = as.vector(bernoulli_sets))
 
# Print head of data
head(sets_data)
#   Set Sample
# 1   1      1
# 2   1      1
# 3   1      1
# 4   1      1
# 5   1      0
# 6   1      1

Each row in the data frame represents a single trial from one of the sets of trials. The Set column identifies which set the trial belongs to, and the Sample column shows the outcome of the trial.

Now we’ll visualize all the sets together.

# Plot multiple sets of Bernoulli trials using ggplot2
ggplot(sets_data, aes(x = factor(Sample), fill = factor(Set))) +
  geom_bar(position = "dodge") +
  labs(
    title = "Multiple Bernoulli Trial Sets",
    x = "Outcome",
    y = "Count",
    fill = "Trial Set"
  ) +
  scale_fill_manual(
    values = c("#1b98e0", "#ff5733", "#33b5ff", "#e9b000", "#c0e2e6")
  ) +
  theme_minimal() +
  theme(
    legend.position = "top"
  )

The plot shows how each set of Bernoulli trials follows the same statistical behavior, but the actual outcomes may differ due to randomness. This highlights the variability inherent in Bernoulli sampling.

Conclusion & Further Resources

In this tutorial, we explored how to simulate and visualize Bernoulli sampling in R using the rbinom() function and ggplot2. We discussed the theoretical foundations, generated Bernoulli trial data, and visualized the results to observe key statistical properties and patterns. Bernoulli sampling is widely used for modeling binary outcomes and plays an important role in fields such as statistics, machine learning, and biology.

If you’re interested in expanding your knowledge of random data generation and statistical modeling in R, you might find the following resources helpful:

• Sampling Theory in Statistics Explained
• How to Simulate a Random Walk in R
• sample Function in R
• Random Numbers in R
• Generate Matrix with i.i.d. Normal Random Variables in R
• Generate Set of Random Integers from Interval in R
• Why & How to Set a Random Seed in R

These articles cover essential techniques for generating and working with random data, which are valuable for building more complex simulations and statistical models.

Please let me know in the comments if you have any questions!