Binomial Distribution in R

Introduction to Binomial Distribution in R

This article describes how to use binomial distributions in R for the few operations involved with probability distributions. Business Analysis makes use of binomial probability for a complex problem. R has numerous built-in Functions for calculating Binomial distributions used in statistical interference. The binomial distribution also known as Bernoulli trials takes two types of success p and failure S. The main goal of the binomial distribution model is they compute the possible probability outcomes by monitoring a specific number of positive possibilities by repeating the process a particular number of times. They should have two possible results (success/ failure), therefore the outcome is dichotomous. The pre-defined mathematical notation is p=success, q=1-p.

There are four functions associated with Binomial distributions. They are dbinom, pbinom, qbinom, rbinom. The formatted syntax is given below:

Syntax

• dbinom(x, size,prob)
• pbinom(x, size,prob)
• qbinom(x, size,prob) or qbinom(x, size,prob , lower_tail,log_p)
• rbinom(x, size,prob)

The function has three arguments: the value x is a vector of quantiles (from 0 to n), size is the number of trails attempts, prob denotes probability for each attempt. Let’s see one by one with an example.

1)dbinom()

It is a density or distribution function. The vector values must be a whole number shouldn’t be a negative number. This function attempts to find a number of success in a no. of trials which are fixed.

A binomial distribution takes size and x values. for example, size=6, the possible x values are 0,1,2,3,4,5,6 which implies P(X=x).

n <- 6; p<- 0.6; x <- 0:n
dbinom(x,n,p)

Output:

Making probability to one

n <- 6; p<- 0.6; x <- 0:n
sum(dbinom(x,n,p))

Output:

Example 1 – Hospital database displays that the patients suffering from cancer, 65% die of it. What will be the probability that of 5 randomly chosen patients out of which 3 will recover?

Here we apply the dbinom function. The probability that 3 will recover using density distribution at all points.

n=5, p=0.65, x=3

dbinom(3, size=5, prob=0.65)

Output:

For x value 0 to 3 :

dbinom(0, size=5, prob=0.65) +
+ dbinom(1, size=5, prob=0.65) +
+ dbinom(2, size=5, prob=0.65) +
+ dbinom(3, size=5, prob=0.65)

Output:

Next, create a sample of 40 papers and incrementing by 2 also creating binomial using dbinom.

a <- seq(0,40,by = 2)
b <- dbinom(a,40,0.4)
plot(a,b)

It produces the following output after executing the above code, The binomial distribution is plotted using plot() function.

Example 2 – Consider a scenario, let’s assume a probability of a student lending a book from a library is 0.7. There are 6 students in the library, what is the probability of 3 of them lending a book?

here P (X=3)

Code:

n=3; p=.7; x=0:n; prob=dbinom(x,n,p);
barplot(prob,names.arg = x,main="Binomial Barplot\n(n=3, p=0.7)",col="lightgreen")

Below Plot shows when p > 0.5, therefore binomial distribution is positively skewed as displayed.

Output:

2)Pbinom()

calculates Cumulative probabilities of binomial or CDF (P(X<=x)).

Example 1:

x <- c(0,2,5,7,8,12,13)
pbinom(x,size=20,prob=.2)

Output:

Example 2: Dravid scores a wicket on 20% of his attempts when he bowls. If he bowls 5 times, what would be the probability that he scores 4 or lesser wicket?

The probability of success is 0.2 here and during 5 attempts we get

pbinom(4, size=5, prob=.2)

Output:

Example 3: 4% of Americans are Black. Find the probability of 2 black students when randomly selecting 6 students from a class of 100 without replacement.

When R: x = 4 R: n = 6 R: p = 0. 0 4

pbinom(4,6,0.04)

Output:-

3)qbinom()

It’s a Quantile Function and does the inverse of the cumulative probability function. The cumulative value matches with a probability value.

Example: How many tails will have a probability of 0.2 when a coin is tossed 61 times.

a <- qbinom(0.2,61,1/2)
print(a)

Output:-

4)rbinom()

It generates random numbers. Different outcomes produce different random output, used in the simulation process.

Example:-

rbinom(30,5,0.5)
rbinom(30,5,0.5)

Output:-

Each time when we execute it gives random results.

rbinom(200,4,0.4)

Output:-

Here we do this by assuming the outcome of 30 coin flips in a single attempt.

rbinom(30,1,0.5)

Output:-

Using barplot:

a<-rbinom(30,1,0.5)
print(a)
barplot(table(a), border=FALSE)

Output:-

To find the mean of success

output <-rbinom(10,size=60,0.3)
mean(output)

Output:-

Conclusion- Binomial Distribution in R

Hence, in this document we have discussed binomial distribution in R. We have simulated using various examples in R studio and R snippets and also described the built-in functions helps in generating binomial calculations. Binomial distribution calculation in R uses statistical calculations. Therefore, a binomial distribution helps in finding probability and random search using a binomial variable.

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