What are the conditions for a distribution to be binomial?

Fixed number of trials, only two possible outcomes per trial, constant probability of success, and independent trials.

Can binomial distribution be used for more than two outcomes?

No, it is strictly for binary outcomes. For more, use multinomial distribution.

How is binomial distribution different from normal distribution?

Binomial is discrete and for a fixed number of trials. Normal is continuous and often used as an approximation for binomial when ( n ) is large and ( p ) is not too close to 0 or 1.

What is the relationship between binomial and Bernoulli distributions?

Binomial describes the number of successes in multiple independent Bernoulli trials.

When do we use binomial approximation to the Poisson distribution?

When ( n ) is large and ( p ) is small such that ( np ) is moderate, binomial can be approximated by Poisson.

How can I quickly calculate binomial probabilities in exams?

Learn to use binomial tables or calculator functions. For small ( n ), manual calculation is feasible.

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JEE Maths

Binomial Distribution

The binomial distribution is a cornerstone concept in probability and statistics, especially relevant for the JEE Maths syllabus. It models the probability of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. Mastery of this topic is essential for scoring high in competitive exams like JEE, where questions often test both conceptual clarity and problem-solving speed.

1.0Definition of Binomial Distribution

A random variable X is said to follow a Binomial Distribution if it represents the number of successes in n independent Bernoulli trials, each with the same probability of success p.

We denote:

X∼B(n,p)

Here:

n = number of trials
p = probability of success
q=1−p = probability of failure

2.0Key Terms in Binomial Distribution

Random Experiment: An experiment whose outcome cannot be predicted with certainty is called a random experiment.
Example: Tossing a coin, rolling a die.
Trial: Each performance of a random experiment is called a trial.
Bernoulli Trial: A Bernoulli trial is a random experiment that results in exactly two possible outcomes: success or failure.

Tossing a coin → Success (Head), Failure (Tail).
Rolling a die → Success (getting a 6), Failure (not getting a 6).

Success and Failure

Probability of success = p
Probability of failure = q=1−p

Also Read: Binomial Theorem

3.0Conditions for Binomial Distribution

The following conditions must be satisfied for a random variable to have a binomial distribution:

There are n independent trials.
Each trial has only two outcomes: success (with probability p) or failure (with probability q).
Probability of success p remains constant for each trial.
Random variable X counts the number of successes.

4.0Binomial Distribution Formula

The probability of getting exactly r successes in n independent trials is given by:

$P (X = r) = (r n) p^{r} q^{n - r}, r = 0, 1, 2, \dots, n$

Where:

$(r n) = \frac{n !}{r ! ( n - r )!}$ is the binomial coefficient.
$p^{r}$ represents the probability of r successes.
$q^{n - r}$ represents the probability of n−r failures.

Example: Probability of getting exactly 3 heads in 5 tosses of a fair coin :

5.0Properties of Binomial Distribution

Discrete Probability Distribution: It takes integer values from 0 to n.
Sum of Probabilities = 1: $\sum_{r = 0}^{n} (r n) p^{r} q^{n - r} = (p + q)^{n} = 1$
Symmetry: Distribution is symmetric when p=q=0.5.
Mode of Distribution: Mode = ⌊(n+1)p⌋.
Additivity: If $X \sim B (n, p) an d Y \sim B (m, p), t h e n X + Y \sim B (n + m, p)$

6.0Mean, Variance, and Standard Deviation

For X∼B(n,p):

Mean (Expectation): E(X)=np
Variance: Var(X)=npq
Standard Deviation (SD): $σ = n pq$

These are very important for JEE probability and statistics problems.

7.0Relation with Bernoulli Trials

A Bernoulli Trial is a single experiment with two outcomes (success or failure).
A Binomial Distribution is the distribution of the number of successes in n independent Bernoulli trials.
If X∼B(n,p), then each trial is Bernoulli with success probability p.

8.0Solved Examples on Binomial Distribution

Example 1: Tossing Coins

Q: A coin is tossed 8 times. What is the probability of getting exactly 5 heads?

Solution:

( n = 8 )
( k = 5 )
( p = 0.5 )

$P (X = 5) = (5 8) (0.5)^{5} (0.5)^{3} = 56 \times 0.03125 \times 0.125 = 0.21875$

Example 2: Multiple Choice Questions

Q: In a quiz with 10 questions, each with 4 options (only one correct), what is the probability of getting exactly 3 correct answers by guessing?

( n = 10 )
( k = 3 )
( p = 0.25 )

$P (X = 3) = (3 10) (0.25)^{3} (0.75)^{7} \approx 120 \times 0.015625 \times 0.133484 = 0.250$

Example 3: Defective Bulbs

Q: A box contains 12 bulbs, 3 of which are defective. If 4 bulbs are chosen at random, what is the probability that exactly one is defective?

Solution:

Total ways to choose 4 bulbs: ( $(4 12)$ )

Ways to choose 1 defective: ( $(1 3)$ )

Ways to choose 3 non-defective: ( $(3 9)$ )

Probability:

$P (X = 1) = \frac{( 1 3 ) ( 3 9 )}{( 4 12 )} = \frac{3 \times 84}{495} = \frac{252}{495} \approx 0.509$