Probability helps students understand uncertainty, prediction, and data-based reasoning, and Chapter 13: Probability is one of the most important and scoring chapters in Class 12 Maths. This chapter builds on basic probability concepts and introduces students to conditional probability, Bayes’ theorem, random variables, and probability distributions, which are widely used in real-life decision-making and statistics.
NCERT Solutions for Class 12 Maths Chapter 13 – Probability are prepared strictly according to the CBSE syllabus and board exam pattern. These solutions include important CBSE board questions asked between 2018 and 2025 and are explained in a clear, step-by-step manner using simple language and proper mathematical presentation. This helps students gain strong conceptual clarity, practise efficiently, and score well in board examinations.
NCERT Solutions For Class 12 Maths Chapter 13 Probability
Q.
Given that E and F are events such that P(E)=0.6,P(F)=0.3and P(E∩F)=0.2,find P(E∣F) and P(F∣E).
Q.
If P (A) =1/2, P (B) = 0, then P (A|B) is
(A) 0 (B) 1/2
(C) not defined (D) 1
Q.
Let A and B be independent events with P(A) = 0.3 and P(B) = 0.4. Find
(i) P(A ∩ B) (ii) P(A ∪ B)
(iii) P(A|B) (iv) P (B|A)
Q.
A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, ‘the number is even,’ and B be the event, ‘the number is red’. Are A and B independent?
Q.
A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not.
Q.
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
Q.
Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.
Q.
If P(A) = 3/5 and P (B) = 1/5, find P (A ∩ B) if A and B are independent events.
Q.
If A and B are events such that P (A|B) = P(B|A), then
(A) A ⊂ B but A ≠ B (B) A = B
(C) A ∩ B = Φ (D) P (A) = P(B)
Q.
Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’, given that ‘at least one die shows a 3’.
Q.
Compute P(A∣B), if P(B) = 0.5and P(A∩B) =0.32.
Q.
A fair die is rolled. Consider events
E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5}
Find
(i) P(E|F) and P (F|E)
(ii) P(E|G) and P(G|E)
(iii) P((E ∪ F)|G) and P((E ∩ F)|G)
Q.
A black and a red dice are rolled.
(a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.
(b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
Q.
Mother, father and son line up at random for a family picture
E: son on one end,
F: father in middle
Determine P (E|F).
Q.
A die is thrown three times,
E: 4 appears on the third toss,
F: 6 and 5 appears respectively on first two tosses
Determine P (E|F).
Q.
Two coins are tossed once, where
(i) E: tail appears on one coin,
F: one coin shows head
(ii) E: no tail appears, F : no head appears
Determine P (E|F).
Q.
IfP(A)=116,P(B)=115and(A∪B=117,find(i)P(A∩B)(ii)P(A∣B)(iii)P(B∣A).
Q.
Evaluate P(A∪B),if 2P(A)=P(B)=135and P(A∣B)=52.
Q.
If P(A)=0.8,P(B)=0.5and P(B∣A)=0.4, find(i)P(A∩B)(ii)P(A∣B)(iii)P(A∪B)
Q.
An experiment succeeds twice as often as it fails. Find the probability that in the next six trials, there will be at least 4 successes.
Q1. Given that E and F are events such that
P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2.
Find P(E | F) and P(F | E).
Answer:
P(E | F) = 2/3 and P(F | E) = 1/3.
Q2. Compute P(A | B), if P(B) = 0.5 and P(A ∩ B) = 0.32.
Answer:
P(A | B) = 0.64.
Q3. If P(A) = 0.8, P(B) = 0.5 and P(B | A) = 0.4, find:
(i) P(A ∩ B)
(ii) P(A | B)
(iii) P(A ∪ B).
Answer:
P(A ∩ B) = 0.32,
P(A | B) = 0.64,
P(A ∪ B) = 0.98.
Q4. If the probability of an event A is 0.45, find the probability of not A.
Answer:
P(not A) = 0.55.
Q5. If P(A ∪ B) = 0.7, P(A) = 0.4 and P(B) = 0.5, find P(A ∩ B).
Answer:
P(A ∩ B) = 0.2.
FAQs: NCERT Solutions for Class 12 Maths Chapter 13 – Probability
Q1. Why is Probability important for Class 12 Maths?
Probability develops logical thinking and analytical skills and carries high weightage in CBSE board exams. It also has strong applications in statistics, economics, and science.
Q2. How many marks are usually asked from Chapter 13 in board exams?
Generally, 8 to 10 marks worth of questions are asked from Probability, often as long-answer questions.
Q3. Are NCERT Solutions enough to score well in Probability?
Yes. NCERT textbook questions and NCERT Solutions are sufficient for CBSE board exams, as most questions are directly based on NCERT exercises and examples.
Q4. Which topics are most important in the Probability chapter?
The most important topics include:
Q5. Why do students find Probability difficult?
Students often struggle with:
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Understanding conditional probability
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Applying Bayes’ theorem correctly
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Avoiding calculation mistakes
Regular practice and clear step-by-step solving help overcome these difficulties.
Q6. Is Probability useful for competitive exams?
Yes. Probability is very important for JEE and other competitive exams, as it frequently appears in questions related to statistics and data interpretation.