Independent events and the multiplication rule: Class 12 practice

What you should know first

• Basic probability

The idea behind independent events and the multiplication rule

Events A and B are independent if P(A ∩ B) = P(A)P(B). P(A ∪ B) = P(A) + P(B) − P(A ∩ B). 'At least one' is 1 − P(neither) = 1 − (1 − P(A))(1 − P(B)).

Step-by-step method

1. Check independence (given or by the product rule).
2. Multiply for 'both'.
3. Use 1 − P(neither) for 'at least one'.

Worked example 1 (Basic)

A and B are independent with P(A) = 2/5, P(B) = 3/5. Find P(A and B).

1. P(A)P(B) = 6/25
2. Answer: 6/25

Worked example 2 (Standard)

A and B are independent with P(A) = 1/4, P(B) = 2/3. Find P(A or B).

1. P(A) + P(B) − P(A)P(B)
2. = 3/4
3. Answer: 3/4

Worked example 3 (Challenge)

Two shooters hit a target with probabilities 3/4 and 1/2 independently. Find P(target is hit).

1. P(both miss) = 1/8
2. P(hit) = 1 − that = 7/8
3. Answer: 7/8

Common mistakes and misconceptions

• Adding probabilities for 'both'
• Assuming events are independent without checking

Practice questions

1. (Basic) A and B are independent with P(A) = 1/4, P(B) = 1/2. Find P(A and B).
2. (Basic) A and B are independent with P(A) = 1/3, P(B) = 2/3. Find P(A and B).
3. (Standard) A and B are independent with P(A) = 1/2, P(B) = 3/5. Find P(A or B).
4. (Standard) A and B are independent with P(A) = 1/2, P(B) = 1/6. Find P(A or B).
5. (Challenge) Two shooters hit a target with probabilities 2/5 and 1/6 independently. Find P(target is hit).
6. (Challenge) Two shooters hit a target with probabilities 3/4 and 3/5 independently. Find P(target is hit).

Answer key with full working

• Q1: P(A)P(B) = 1/8 → Answer: 1/8
• Q2: P(A)P(B) = 2/9 → Answer: 2/9
• Q3: P(A) + P(B) − P(A)P(B) → = 4/5 → Answer: 4/5
• Q4: P(A) + P(B) − P(A)P(B) → = 7/12 → Answer: 7/12
• Q5: P(both miss) = 1/2 → P(hit) = 1 − that = 1/2 → Answer: 1/2
• Q6: P(both miss) = 1/10 → P(hit) = 1 − that = 9/10 → Answer: 9/10

Syllabus note

This skill is listed under "Probability" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

• Conditional probability and Bayes' theorem
• Binomial distribution
• Mean and variance of a random variable
• Law of total probability
• Derivatives of polynomials and slope of a tangent
• Definite integrals of polynomials
• Evaluating 2×2 and 3×3 determinants
• Matrix multiplication
• All Class 12 practice

Frequently asked questions

Which chapter is independent events and the multiplication rule in?

"Probability", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.