Binomial Distribution: Overview, Questions, Preparation

Binomial distribution in simplest terms is the probability of having only two possible outcomes – success or failure. The binomial distribution is also termed as Bernoulli distribution. This forms one of the most important bases of strategic calculations.

Now, let’s understand this in the form of a mathematical equation.

When a dice is thrown 10 times, then the outcome can be 6 different outcomes. Thus, the probability of getting 6 in a dice throw can be 1/6.
It can be further written as, n = 10, p = 1/6

A simple binomial distribution is all about success or failure. Let the success be written as p and failure be written as q. Thus, the probability of getting success will be 1/p, and the probability of getting failure will be 1/q.

It can also be written as, p = q – 1 

Mean and variance of Binomial Distribution

If 's' is the probability of success and 'q' is the probability of failure in a binomial distribution, then the number of success in 'n' trials means the mean value of the binomial distribution is E(X) = μ = ns.

The variance of binomial distribution is V(X) = σ2 = nsf.

n – Number of Experiments

x – Random variable – 0,1,2,3,4

p – Probability of success in a single experiment

q – Probability of failure in a single experiment, which is also written as q = 1- p

Thus, the binomial distribution formula is written as –

P(x:n,p) = n!/[x!(n-x)!].p^x.(q)^n-x

The binomial distribution is a regular topic in Class XII. This is a high scoring area from where some questions are definitely expected every year. This topic falls under the probability chapter.

A total of 10 marks is allocated to probability chapters. You can expect a mix of MCQ and short answer type questions from this topic. There can be around 3-4 questions from game theory topics. 

Question 1. ‘n’ is the number of trials and ‘p’ is the probability of success, find out the mean value?  

Solution: In case of a discrete probability function, the mean value or the expected value is,

Mean (μ) = np

For Binomial Distribution P(x)=nCx px q(n-x), substitute in above equation and solve to get
µ = np.

Question 2. In a dice throw, n - number of trials, p - Probability of success, q - the probability of failure, x - random variable. Find out the probability that X takes the value of x? 

Solution: It is the formula for Binomial Distribution that is asked here which is given by P(X = x) = nCx px q(n-x).

Question 3. A large collection of tires has 3% defective tires. Suppose one choose tires from this collection until he/she obtains 4 non-defective tires. Then the total number of defective tires drawn in this process has a Negative Binomial distribution with ______

Solution:  r = Total number of success = 4, p = Probability of success

= 1 − 0.03 = 0.97.

Q: A Negative Binomial(r, p) random variable can be expressed as a sum of r Geometric(p) random variables. This statement is.

Q: How Binomial Theorem is used in Algebra?

Q: If success is represented as p and failure as q, then how is p written :

Q: Give an example of a binomial distribution

Q: What are the four conditions of a binomial distribution?