Introduction to the binomial distribution



The binomial distribution is a discrete probability distribution often used in real-life situations where binary outcomes take place. Success-failure, Pass-fail, and Yes-no are some of the binary outcomes.

This is a complete guide for a beginner who wanted learn the binomial distribution. The following terms you will learn in this article.

  1. Binomial distribution formula
  2. Use of technology to find binomial probabilities.
  3. Microsoft Excel
  4. TI83/TI84 calculators

 

Suppose X be a random variable, that counts a number of successes among a total of n trials. Consider the following conditions.

  1. The total number of trials (n) is fixed.
  2. Each trial is independent of the others.
  3. Each trial is a binary trial, that is, for each trial,      there are only two outcomes. One of these outcomes is success and another is a failure.
  4. The probability of success, p, is the same for each trial.

If X satisfies all these conditions then X follows a binomial distribution with the number of trials (n) and probability of success (p).


Binomial distribution formula

Let a random variable X follows a binomial distribution with a number of trials n and probability of success p then the probability mass function (pmf) of random variable X (or binomial distribution formula) is given by,



Where

n = Number of trials

x = Desired number of successes

p = Probability of getting success

1-p = Probability of getting failure


Technology use:

The binomial distribution is a discrete distribution. As the number of trials (n) increases it is quite hard to compute binomial probabilities using the formula. Technologies make this simple. Let us understand how one can use technologies to find the binomial probabilities as follows.


Excel:

Microsoft Excel is the tool that is easily available on almost all laptops and computers nowadays. This tool is specially used to make calculations simpler.

There is a function in Microsoft excel called =BINOM.DIST(s, n, p, #) to find binomial probabilities. Within this function, you need to plug the values of the desired number of successes (s), the desired number of trials (n), and desired probability of success (p). In addition, # in the function denotes logical 0(FALSE) or 1(TRUE) value. Use logical 0(FALSE) if you need to find the probability at single point X=x, that is, the probabilities of kind


P(X=x)


Use logical 1(TRUE) if you need to find the cumulative probability at X=x. The cumulative probability at X=x means the sum of all the probabilities at x and below x, that is, the probabilities of kind


P(X ≤ x) = P(X=0) + P(X=1) + ... + P(X=x)


There are three ways to call the function in an excel sheet.

  1. Simply select any cell in an excel sheet then type =BINOM.DIST(s, n, p, #) then hit ENTER button.
  2. Click on the symbol of “Insert Function” present in the formula bar. 

    The new window of Insert Function will open. Then search for the function =BINOM.DIST(s, n, p, #) in this window. 

  3. Go to the Formulas menu at the top. Then click on the Insert Function tab in this menu. 

         The new window of Insert Function will open.  Then search for the function =BINOM.DIST(s, n, p, #) in this window. 

 

Illustration

Suppose number of successes (s) = 2, number of trials (n) =10, and probability of success (p) = 0.25. Then the excel function =BINOM.DIST(s, n, p, #) gives the binomial probability at X= 2 is, 0.281568.


 

While the cumulative probability at X = 2 is 0.525593.

 


 

TI83/TI84 calculator:

TI83/TI84 calculator is a modern scientific calculator wherein various mathematical, as well as statistical calculations, are possible. One can use the calculator to graph the functions but there are some limitations. Let us not get into more details and focus on how to calculate binomial probabilities using the TI83/TI84 calculator.

Follow the path below.

2ND+VARS >>> DISTR >>>

 


When you scroll down under DISRT you will find two functions, one is A: binompdf() and another is B: binomcdf(). 

Use the function A: binompdf() if you need to find the probability at single point X=x, that is, the probabilities of kind


P(X=x)


Use the function B: binomcdf() if you need to find the cumulative probability at X=x. The cumulative probability at X=x means the sum of all the probabilities at x and below x, that is, the probabilities of kind


P(X≤x) = P(X=0) + P(X=1) + ... + P(X=x)




In both functions A: binompdf() and B: binomcdf() you need to plug the values of the number of trials (trials), probability of success (p), and the number of successes (x value). Then after hitting the ENTER button twice you will get the desired binomial probability.

 

Illustration

Suppose number of successes (s) = 2, number of trials (n) =10, and probability of success (p) = 0.25. Then the TI83/TI84 function A: binompdf() gives the binomial probability at X= 2.

 


While the function B: binomcdf() gives cumulative probability at X = 2.


This gives P(X=2)=0.2815675735 and P(X≤2)=0.525592804.