Binomial Distribution in R Programming

Introduction

The Binomial Distribution model is an essential part of statistical analysis that provides powerful insights into probabilities and events. It is a discrete probability distribution of the number of successes in independent experiments. We can easily manipulate, analyze, and visualize these distributions by harnessing the power of R, a popular language among statisticians and data scientists. This article will delve into the theoretical underpinnings of the Binomial Distribution and its applications and illustrate how to leverage R programming for implementing and visualizing it.

What is Binomial Distribution?

A binomial distribution represents a probability distribution that describes the outcome of a fixed number of independent and identically distributed Bernoulli trials, each with a constant probability of success. In simpler terms, it's the probability of a SUCCESS or a FAILURE outcome in an experiment or survey that is repeated multiple times.

The binomial is a distribution type with two possible outcomes (the prefix 'bi' represents two or twice). For example, a coin toss has two possible outcomes: heads or tails.

Key properties of a binomial distribution are:

1. The observations' number or trials is fixed. In other words, you can only figure out the probability of something happening if you do it a certain number of times.
2. Each observation or trial is independent. In other words, none of your trials affect the probability of the next trial.

These characteristics make the binomial distribution suitable for a wide range of real-world scenarios and problems.

Here are some real-life scenarios where the binomial distribution is applicable:

1. Coin Tossing: Suppose you toss a fair coin ten times. The binomial distribution can be used to determine the probability of getting a certain number of heads (or tails). Each coin toss is an independent event, the probability of getting a head or tail is the same for each toss (0.5), and you are conducting a fixed number of trials (10 coin tosses).

2. Quality Control in Manufacturing: A factory produces items and each item may be defective or not defective. If you randomly select a certain number of items (fixed number of trials), the binomial distribution can be used to calculate the probability of finding a specific number of defective items. Each selection is an independent event and the probability of selecting a defective item is the same for each selection.

3. Medical Trials: Suppose a drug has a 70% chance of curing a certain disease. If the drug is given to 50 patients (fixed number of trials), the binomial distribution can help estimate the probability of the drug curing a certain number of patients. Each patient's outcome is independent of the others and the probability of success (curing the disease) is the same for each trial.

4. Survey Sampling: If you're conducting a survey and you know that 60% of a population will choose option A (based on past data or a larger sample), a binomial distribution can help you determine the probability that a certain number out of a smaller sample will choose option A. Each survey response is an independent event, the probability of each person choosing option A is the same, and you're surveying a fixed number of people.

5. Sports: If a basketball player makes a free throw 80% of the time, you can use the binomial distribution to calculate the probability of that player making a certain number of free throws out of a fixed number of attempts. Each free throw is an independent event, and the probability of success is the same for each shot.

These examples demonstrate the wide applicability of the binomial distribution in different domains.

Formula of Binomial Distribution

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. A Bernoulli trial is an experiment or process that results in a binary outcome, often termed "success" or "failure".

The probability mass function (PMF) of the binomial distribution is given by:

[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ]

Where:

• ( P(X = k) ) is the probability of observing ( k ) successes in ( n ) trials.
• ( \binom{n}{k} ) is the binomial coefficient, representing the number of ways to choose ( k ) successes from ( n ) trials. It's calculated as ( \frac{n!}{k!(n-k)!} ), where ( ! ) denotes factorial.
• ( p ) is the probability of success on a single trial.
• ( 1-p ) is the probability of failure on a single trial.
• ( n ) is the number of trials.
• ( k ) is the number of successes.

In this formula, \( k \) can take any value from 0 to ( n ).

dbinom()

The dbinom() function in R is a powerful tool for computing a binomial distribution's probability density (mass) function. It allows users to calculate the probability of obtaining a specific number of "successes" in a fixed number of Bernoulli trials, given a certain probability of success. By leveraging dbinom(), users can enhance their statistical analysis, offering a clearer understanding of binomial distributions in practical scenarios.

Here's how you can use the dbinom() function in R:

The output for this code might look something like this:

This suggests that the probability of getting exactly 3 heads in 5 coin tosses is 0.3125, assuming a fair coin (where the probability of getting a head in each toss is 0.5).

Here's another example using pbinom(), which gives us the cumulative probability distribution, i.e., the probability of getting 'x' successes or fewer:

The output for this code might look like this:

This indicates that the probability of getting 5 or fewer successes in 20 trials, with the probability of success on each trial being 0.25, is approximately 0.882. In these code snippets, 'x' is the number of successes we're interested in, 'size' is the number of trials, and 'prob' is the probability of success on each trial.

pbinom()

The pbinom() function in R calculates the cumulative probability of a binomial distribution. This function is incredibly helpful when we need to compute the probability of having a certain number of successes or fewer in a given number of independent trials.

Here's an example of how to use pbinom():

The 'q' parameter is the number of successes we're interested in, 'size' represents the number of trials, and 'prob' is the probability of success on each trial.

Running this code snippet, you might see an output like:

This suggests that the probability of getting 4 heads or fewer in 10 coin tosses is approximately 0.623, assuming a fair coin (where the probability of getting a head in each toss is 0.5).

qbinom()

The qbinom() function in R provides the inverse of the pbinom() function. It returns the smallest number of successes in a set of Bernoulli trials for which the cumulative probability is greater than or equal to a specified probability level. In other words, qbinom() is used for a binomial distribution to find the quantile function, or the number of successes at a given percentile.

Here's an example of how to use qbinom():

In this code, 'p' is the percentile we're interested in (expressed as a probability), 'size' is the number of trials, and 'prob' is the probability of success on each trial.

Running this code snippet, you might see an output like:

This suggests that in 10 coin tosses, we'd expect to see 6 or fewer heads 70% of the time, assuming a fair coin (where the probability of getting a head in each toss is 0.5).

rbinom()

R's rbinom() function allows us to generate random numbers following a binomial distribution. This can be incredibly useful in various situations, such as simulating experiments, bootstrapping, or validating statistical models.

Here's how you can use rbinom():

In this code, 'n' is the number of random numbers we want to generate, 'size' is the number of trials, and 'prob' is the probability of success on each trial.

Running this code snippet, you might see an output like:

These are the number of successes in each of the 100 experiments. For instance, in the first experiment, we got 5 heads, in the second experiment, we got 6 heads, and so on.