Binomial Distribution

What is Binomial Distribution?

Bi means two, Binomial Distribution is a type of probability distribution that has exactly two outcomes. For example, the tossing of a coin has exactly two possible outcomes i.e. a head or a tail. It deals with the probability of success (p) or failure (q) of an experiment or when the experiment is carried out multiple times (n = number of times the experiment is repeated).

Criteria for Binomial Distribution

Criteria for binomial distribution are as follows:

• The number of times the experiment is carried out i.e. n is fixed.
• Each of the observations carried out is independent of one another i.e. all the n observations are independent.
• The probability of success = 1- The probability of failure, i.e., p = 1-q and the probability of success is constant i.e. exactly the same for all the n observations.

Once it is confirmed that the distribution follows and obeys all the criteria and that the distribution is binomial, the binomial distribution formula can be used to compute the probability.

Binomial Distribution Formula

f(x) = ⁿC_x p^x q^n-x  = ⁿC_x p^x (1-p)^n-x

x = total number of successes
p = probability of success
q = probability of failure = 1-p
n = number of trials/ observations

The binomial distribution has the following properties:

1. The mean of the distribution (μx) is equal to n * P .
2. The variance (σ2x) is n * P * ( 1 - P ).
3. The standard deviation (σx) is sqrt[ n * P * ( 1 - P ) ]

Example

80% of the interns working with a company are male. If 9 interns are randomly selected, find the probability that exactly 6 are men.

Solution:
Using formula,

Here, n is the number of randomly selected interns = 9
X is the number you are asked to find the probability for = 6

Substituting the values in the first part of the formula,
n! / (n – X)!  X! = 9! / ((9 – 6)! × 6!) = 84

Now, p = probability of success = 80%
Q = probability of failure = 20%

Working the second part of the formula, p^X

= 0.8^6

= 0.262144

And, q^(n – X)

= 0.2^(9-6)

= 0.008

Putting together the values, the final answer is 0.176.

Frequently Asked Questions

What is binomial distribution?

Binomial Distribution is a type of probability distribution that has exactly two outcomes. For example, the tossing of a coin has exactly two possible outcomes i.e. a head or a tail.

What are the properties of binomial distribution?

●   The number of times the experiment is carried out i.e. n is fixed.
●   Each of the observations carried out is independent of one another i.e. all the n observations are independent.
●   The probability of success = 1- The probability of failure, i.e., p = 1-q and the probability of success is constant i.e. exactly the same for all the n observations.

Why is it called binomial distribution?

Swiss mathematician Jakob Bernoulli, in a proof published posthumously in 1713, determined that the probability of k such outcomes in n repetitions is equal to the kth term (where k starts with 0) in the expansion of the binomial expression (p + q)ⁿ, where q = 1 − p. Hence the name binomial distribution.

Conclusion

In this article, we have extensively discussed Bernoulli trial and Binomial Distribution. We hope that this blog has helped you enhance your knowledge and if you wish to learn more, check out our Code360 Blog site and visit our Library. Do upvote our blog to help others grow.

Happy Learning!