Binomial Distribution – Solved Problems
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. This topic is fundamental in statistics, probability theory, data science, and competitive exams.
where q = 1 − p
Problem 1: Basic Binomial Probability
A coin is tossed 5 times. Find the probability of getting exactly 3 heads.
Solution:
Number of trials, n = 5 Probability of head, p = 1/2 Probability of tail, q = 1/2 Number of successes, x = 3
Answer: Probability = 5/16
Problem 2: Manufacturing Defects
The probability that a product is defective is 0.1. If 6 products are selected at random, find the probability that exactly 2 are defective.
Solution:
n = 6, x = 2 p = 0.1, q = 0.9
Answer: Probability ≈ 0.098
Problem 3: Exam Pass Probability
The probability that a student passes an exam is 0.7. If 4 students appear independently, find the probability that at least 3 students pass.
Solution:
“At least 3 pass” means X = 3 or X = 4
For X = 3:
For X = 4:
Answer: Probability ≈ 0.652
Problem 4: Mean and Variance
For a binomial distribution with n = 10 and p = 0.4, find the mean and variance.
Solution:
q = 1 − 0.4 = 0.6
Mean = 10 × 0.4 = 4 Variance = 10 × 0.4 × 0.6 = 2.4
Answer: Mean = 4, Variance = 2.4
Problem 5: Multiple Choice Questions
A multiple-choice question has 4 options, only one of which is correct. If a student guesses the answer to 5 such questions randomly, find the probability that exactly 2 answers are correct.
Solution:
p = 1/4, q = 3/4, n = 5, x = 2
Answer: Probability = 135/512
Problem 6: At Most Probability
A biased coin has probability of heads equal to 0.6. If the coin is tossed 3 times, find the probability of getting at most one head.
Solution:
“At most one head” means X = 0 or X = 1
Answer: Probability = 0.352
Problem 7: Real-Life Interpretation
The probability that a machine produces a defective item is 0.05. If the machine produces 8 items, what is the probability that none of them is defective?
Solution:
p = 0.05, q = 0.95, n = 8
Answer: Probability ≈ 0.663
Conclusion: Binomial distribution helps model real-life success–failure experiments. Understanding its formula, assumptions, and applications is essential for probability, statistics, and data analysis.