Class 12 Maths · Chapter 13
Unit: Probability
Tentative Study Share: 8.0 Marks
Chapter 13: Probability — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Probability.
NCERT Exercises in Chapter 13: Probability
📘 NCERT Concept Notes — Probability
01 · Conditioning changes the sample space
After learning that F occurred, keep only outcomes in F and renormalise their probabilities. Do not reverse the condition: P(E|F) and P(F|E) generally differ.
02 · Multiply along a path
The second probability in a sequence is conditional on the first event. Independence allows that conditional to be replaced by an unconditional probability, but cannot be assumed from wording alone.
03 · A partition accounts for every route
A partition splits the sample space into pairwise disjoint, exhaustive events. Total probability adds the joint probabilities of all routes leading to the observed event.
04 · Bayes reverses the question
A prior describes a route before the observation. A likelihood describes the observation given that route. A posterior updates the route probability after the observation. Check the denominator contains every route.
⚡ Key Formulas & Results — Probability
Conditional probability
What probability is in the denominator of P(E|F)?
\(P(E\mid F)=\frac{P(E\cap F)}{P(F)}\)
Complements after conditioning
What is the complement rule inside the conditioned space?
\(P(E^c\mid F)=1-P(E\mid F)\)
Conditional union
What is the union formula with a common condition?
\(P(E\cup F\mid G)=P(E\mid G)+P(F\mid G)-P(E\cap F\mid G)\)
Multiplication theorem
How do you recover a joint probability from a conditional?
\(P(E\cap F)=P(E)P(F\mid E)\)
Derive the multiplication theorem
How does the conditional definition rearrange?
\(P(E\cap F)=P(F)P(E\mid F)\)
Independence
Which equality defines independence even when a marginal is zero?
\(P(E\cap F)=P(E)P(F)\)
Disjoint is different from independent
Can two disjoint events of positive probability be independent?
\(P(E\cap F)=0\ne P(E)P(F)\)
Partition
What two conditions make E₁,…,Eₙ a partition?
\(E_i\cap E_j=\varnothing\ (i\ne j),\quad\bigcup_iE_i=S\)
Derive total probability
Why is P(A) the weighted sum of its conditional probabilities?
\(P(A)=\sum_iP(E_i)P(A\mid E_i)\)
Bayes formula
How do you find the probability of route Eᵢ after observing A?
\(P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum_jP(E_j)P(A\mid E_j)}\)
🎯 CBSE 2026 Board PYQs from Probability
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q19
Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q31
Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q37
In an online jackpot, there is one first prize of ₹3,00,000, two second prizes of ₹2,00,000 each and…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q20
Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q29
Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q36
In an online jackpot, there is one first prize of ₹3,00,000, two second prizes of ₹2,00,000 each and…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q19
Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q27
Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q38
In an online jackpot, there is one first prize of ₹3,00,000, two second prizes of ₹2,00,000 each and…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q18
For events A and B with P(A)≠0 and P(B)≠1, find P(A′ | B′).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q31
On a stormy day a sniper is three times as likely to hit as to miss with each shot. For two shots, find…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q36
In a group of 50 smokers, 30 are male. Lung complications occur in 170 per 1000 male smokers and 120…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q11
For events A and B with P(A)≠0 and P(B)≠1, find P(A′ | B′).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q29
On a stormy day a sniper is three times as likely to hit as to miss with each shot. For two shots, find…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q38
In a group of 50 smokers, 30 are male. Lung complications occur in 170 per 1000 male smokers and 120…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q4
For events A and B with P(A)≠0 and P(B)≠1, find P(A′ | B′).
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q26
On a stormy day a sniper is three times as likely to hit as to miss with each shot. For two shots, find…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/3 · Q37
In a group of 50 smokers, 30 are male. Lung complications occur in 170 per 1000 male smokers and 120…
📝 Chapterwise Practice Questions — Probability (132 Questions)
Class 12 Probability · Mcq Single Correct
Given P(A)=4/5 and P(A∩B)=7/10, find P(B|A).
Class 12 Probability · Mcq Single Correct
Five independent fair coins are tossed. Find the probability of at least one head.
Class 12 Probability · Mcq Single Correct
Given P(A|B)=0.3, P(A)=0.4, P(B)=0.8, find P(B|A).
Class 12 Probability · Mcq Single Correct
Two independent fair dice have sum 9. Find the conditional probability that at least one die shows 4.
Class 12 Probability · Mcq Single Correct
Choose an ordered pair (a,b) uniformly with distinct a,b from {1,2,3,4,5}. Find P(a/b is an integer).
Class 12 Probability · Mcq Single Correct
Given P(A)=0.4, P(B)=0.3 and P(A∪B)=0.5, find P(B′∩A).
Class 12 Probability · Mcq Single Correct
A bag contains 3 white, 4 black and 2 red balls. Draw 2 uniformly without replacement. Find P(both…
Class 12 Probability · Mcq Single Correct
Given P(B)=3/5, P(A|B)=1/2 and P(A∪B)=4/5, find P(A).
Class 12 Probability · Mcq Single Correct
A,B,C each fire once independently at a target, with hit probabilities 0.4,0.3,0.2. Find P(exactly two…
Class 12 Probability · Mcq Single Correct
A family has three children, independently equally likely to be boys or girls. Given at least one girl…
Class 12 Probability · Mcq Single Correct
Roll a fair die and independently select a uniform card from 52 cards. Find P(even die and spade).
Class 12 Probability · Mcq Single Correct
A box has 3 orange, 3 green and 2 blue balls. Draw 3 uniformly without replacement. Find P(exactly 2…
Class 12 Probability · Mcq Single Correct
Of 8 batteries, 3 are dead. Select 2 uniformly without replacement. Find P(both dead).
Class 12 Probability · Mcq Single Correct
Two independent fair dice are rolled. Given their sum is less than 6, find P(sum is 3).
Class 12 Probability · Mcq Single Correct
Draw two cards independently with replacement from a shuffled 52-card deck. Find P(both queens).
Class 12 Probability · Mcq Single Correct
A bag has 5 red and 3 blue balls. Draw 3 uniformly without replacement. Find P(exactly one red).
Class 12 Probability · Mcq Single Correct
A random variable has values 2,3,4,5 with probabilities 5/k,7/k,9/k,11/k. Find k.
Class 12 Probability · Mcq Single Correct
Two independent fair dice are rolled. A is “first die is 6” and B is “second die is 2”. Find P(A∩B).
Class 12 Probability · Mcq Single Correct
In a college, 30% fail Physics, 25% fail Mathematics, and 10% fail both. Given a uniform student failed…
Class 12 Probability · Mcq Single Correct
A,B solve a problem independently with probabilities 1/3,1/4. Conditional on both being wrong, their…
Class 12 Probability · Mcq Single Correct
Choose uniformly one function from {1,…,10} into itself. Find P(the function is injective).
Class 12 Probability · Mcq Single Correct
If two events are independent, which of the following must hold?
Class 12 Probability · Mcq Single Correct
Given P(A)=1/2, P(B)=1/3 and P(A|B)=1/4, find P(A′∩B′).
Class 12 Probability · Assertion Reason
Assertion (A): For two independent fair coins, P(two heads|at least one head)=1/3. Reason (R): For…