Probability — Class 12 MCQs with Answers

Probability — important questions & MCQs with answers (Class 12 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   P(A|B) =

   A.P(A)/P(B)

   B.P(A∩B)/P(B)✓ Correct

   C.P(A∪B)/P(B)

   D.P(A)·P(B)

   Answer: B. P(A∩B)/P(B)

   Explanation: Conditional probability restricts the sample space to B, so P(A|B) = P(A∩B)/P(B) — dividing by P(B), not by P(A∪B), which double-counts the union.

2. Q2Easy

   Sample space S:

   A.Set of all possible outcomes✓ Correct

   B.Single outcome

   C.Probability

   D.Just events

   Answer: A. Set of all possible outcomes

   Explanation: The sample space S is the complete set of every possible outcome of an experiment — a single outcome is just one sample point, not the whole space.

3. Q3Easy

   An event is:

   A.Probability

   B.Subset of sample space✓ Correct

   C.Random number

   D.Outcome

   Answer: B. Subset of sample space

   Explanation: An event is any subset of the sample space, possibly containing one outcome, several, or none — it is not itself a probability or a single fixed outcome.

4. Q4Easy

   P(E) lies in:

   A.[0, ∞)

   B.[−1, 1]

   C.[0, 1]✓ Correct

   D.R

   Answer: C. [0, 1]

   Explanation: Probability is scaled between 0 (impossible) and 1 (certain), so P(E) always lies in [0,1] — never negative and never above 1, unlike [−1,1] or all of R.

5. Q5Easy

   Events A and B are independent if:

   A.Always true

   B.P(A|B) = 0

   C.P(A) + P(B) = 1

   D.P(A ∩ B) = P(A) · P(B)✓ Correct

   Answer: D. P(A ∩ B) = P(A) · P(B)

   Explanation: Two events are independent exactly when P(A∩B) = P(A)·P(B); P(A|B) = 0 would instead mean they're mutually exclusive, the opposite relationship.

6. Q6Easy

   Mutually exclusive events:

   A.P(A ∪ B) = 0

   B.P(A) = P(B)

   C.P(A ∩ B) = 0✓ Correct

   D.Always overlap

   Answer: C. P(A ∩ B) = 0

   Explanation: Mutually exclusive events share no outcomes, so they can never occur together: P(A∩B) = 0. This says nothing about P(A∪B), which need not be 0.

7. Q7Medium

   Mean (expected value) of random variable X:

   A.Σ xᵢ · P(xᵢ)✓ Correct

   B.Σ xᵢ

   C.Σ P(xᵢ)

   D.max xᵢ

   Answer: A. Σ xᵢ · P(xᵢ)

   Explanation: The mean of a random variable weights each value by its own probability: E(X) = Σxᵢ·P(xᵢ) — not a plain sum Σxᵢ or the count Σ P(xᵢ), which just totals to 1.

8. Q8Medium

   P(getting an even number on a fair die):

   A.1/6

   B.1/3

   C.2/3

   D.1/2✓ Correct

   Answer: D. 1/2

   Explanation: A fair die has three even faces — 2, 4, 6 — out of six total, so P(even) = 3/6 = 1/2, not 1/3, which would undercount the favourable faces.

9. Q9Medium

   For above values, are A and B independent?

   A.Always

   B.No

   C.Cannot determine

   D.Yes (P(A|B) = P(A))✓ Correct

   Answer: D. Yes (P(A|B) = P(A))

   Explanation: Since P(A|B) = P(A∩B)/P(B) = 0.2/0.5 = 0.4 equals P(A) = 0.4, knowing B occurred doesn't change A's chance, so the events are independent by definition.

10. Q10Medium

    Bayes' theorem reverses:

    A.Sum into product

    B.Conditional probability direction (P(A|B) ↔ P(B|A))✓ Correct

    C.Mean and variance

    D.Random

    Answer: B. Conditional probability direction (P(A|B) ↔ P(B|A))

    Explanation: Bayes' theorem converts P(B|A) into P(A|B) by combining it with the prior P(A) and dividing by P(B) — it reverses which event is the 'given' one, not sums into products.

11. Q11Medium

    E(X) = ?

    A.P(X = x)

    B.Σ x

    C.Σ x · P(X = x)✓ Correct

    D.Σ x²

    Answer: C. Σ x · P(X = x)

    Explanation: Expected value weights each outcome by how likely it is: E(X) = Σx·P(X=x) — a plain sum Σx or Σ P(x) ignores the probabilities, or the values, entirely.

12. Q12Hard

    Binomial distribution P(X=k):

    A.pᵏ

    B.C(n,k)·pᵏ·(1−p)^(n−k)✓ Correct

    C.n!·k

    D.k/n

    Answer: B. C(n,k)·pᵏ·(1−p)^(n−k)

    Explanation: Binomial probability counts every arrangement of k successes among n trials: P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ — dropping the C(n,k) or the failure term (1−p)ⁿ⁻ᵏ undercounts the outcomes.

13. Q13Hard

    P(at most 2 heads in 4 coin tosses):

    A.5/16

    B.11/16✓ Correct

    C.1/2

    D.3/4

    Answer: B. 11/16

    Explanation: 'At most 2 heads' sums three cases: P(0)+P(1)+P(2) = C(4,0)/16 + C(4,1)/16 + C(4,2)/16 = 1/16 + 4/16 + 6/16 = 11/16.

14. Q14Hard

    For 3 mutually independent events with probabilities 0.5, 0.4, 0.3, P(all three occur):

    A.0.06✓ Correct

    B.0.12

    C.0.05

    D.0.5

    Answer: A. 0.06

    Explanation: For mutually independent events, the joint probability is a straight product: P(A∩B∩C) = 0.5 × 0.4 × 0.3 = 0.06 — no overlap subtraction is needed since independence already accounts for it.

Probability — FAQs

More Class 12 Mathematics chapters

• 01Relations and Functions
• 02Inverse Trigonometric Functions
• 03Matrices
• 04Determinants
• 05Continuity and Differentiability
• 06Application of Derivatives

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