Law of total probability: Class 12 practice

If events E₁, E₂, … partition the sample space, P(A) = Σ P(Eᵢ)P(A|Eᵢ). Read the method, follow the three worked examples, then try the six questions.

What you should know first

Conditional probability

The idea behind law of total probability

If events E₁, E₂, … partition the sample space, P(A) = Σ P(Eᵢ)P(A|Eᵢ). Use it when an outcome can happen through several routes (bags, machines, factories).

Step-by-step method

List each route Eᵢ with its probability.
Multiply by P(A | route).
Add the products.

Worked example 1 (Basic)

Machines A, B make 2/5, 3/5 of the output with defect rates 7%, 9%. Find P(a random item is defective).

2/5 × 7/100 + 3/5 × 9/100
= 41/500
Answer: 41/500

Worked example 2 (Standard)

Machines A, B, C make 1/3, 1/3, 1/3 of the output with defect rates 5%, 2%, 1%. Find P(a random item is defective).

1/3 × 5/100 + 1/3 × 2/100 + 1/3 × 1/100
= 2/75
Answer: 2/75

Worked example 3 (Challenge)

Machines A, B, C, make 1/8, 5/16, 1/4, 5/16 of the output with defect rates 5%, 7%, 3%, 2%. Find P(a random item is defective).

1/8 × 5/100 + 5/16 × 7/100 + 1/4 × 3/100 + 5/16 × 2/100
= 67/1600
Answer: 67/1600

Common mistakes and misconceptions

Adding conditional probabilities without weights
Weights not summing to 1

Practice questions

(Basic) Machines A, B make 1/3, 2/3 of the output with defect rates 2%, 1%. Find P(a random item is defective).
(Basic) Machines A, B make 3/8, 5/8 of the output with defect rates 5%, 9%. Find P(a random item is defective).
(Standard) Machines A, B, C make 2/11, 4/11, 5/11 of the output with defect rates 5%, 1%, 7%. Find P(a random item is defective).
(Standard) Machines A, B, C make 2/5, 2/5, 1/5 of the output with defect rates 4%, 1%, 5%. Find P(a random item is defective).
(Challenge) Machines A, B, C, make 1/3, 1/15, 4/15, 1/3 of the output with defect rates 6%, 8%, 6%, 6%. Find P(a random item is defective).
(Challenge) Machines A, B, C, make 2/7, 3/14, 1/7, 5/14 of the output with defect rates 3%, 4%, 1%, 8%. Find P(a random item is defective).

Answer key with full working

Q1: 1/3 × 2/100 + 2/3 × 1/100 → = 1/75 → Answer: 1/75
Q2: 3/8 × 5/100 + 5/8 × 9/100 → = 3/40 → Answer: 3/40
Q3: 2/11 × 5/100 + 4/11 × 1/100 + 5/11 × 7/100 → = 49/1100 → Answer: 49/1100
Q4: 2/5 × 4/100 + 2/5 × 1/100 + 1/5 × 5/100 → = 3/100 → Answer: 3/100
Q5: 1/3 × 6/100 + 1/15 × 8/100 + 4/15 × 6/100 + 1/3 × 6/100 → = 23/375 → Answer: 23/375
Q6: 2/7 × 3/100 + 3/14 × 4/100 + 1/7 × 1/100 + 5/14 × 8/100 → = 33/700 → Answer: 33/700

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

Frequently asked questions

Which chapter is law of total probability 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.

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