Class 12 Probability Part 2 | Multiplication Theorem & Independent Events | NCERT | Faiz Sir

Описание: Probability Class 12 Chapter 13 | Part 2 | NCERT | CBSE | Infinix Classes | By Faiz Sir

Is video me hum Class 12 Maths Chapter 13 – Probability ka Part 2 detail me cover kar rahe hain.
Is part me Multiplication Theorem on Probability, Conditional Probability, aur Independent Events ko step-by-step examples, proofs aur important board questions ke saath samjhaya gaya hai.

📌 Topics Covered in This Video:

Multiplication Theorem on Probability

Probability without Replacement Problems

Independent Events (Definitions & Conditions)

Checking Independence of Events

Real-Life Probability Problems

Important NCERT Examples & Board Questions

MCQs & Competitive Level Questions

🎯 Useful for:

CBSE Class 12 Board Exams 2025

NCERT Exam Preparation

Competitive Exams

Concept Clarity in Probability

📚 Learn Maths Conceptually with Infinix Classes — Faiz Sir
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PROBABILITY | Class - 12 | Chapter – 13 | NCERT | By : FAIZ SIR
Multiplication Theorem on Probability:

Example: An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black?

Example: Three cards are drawn successively, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and the third card drawn is an ace?

Independent Events:

Definition 2: Two events E and F are said to be independent, if
P(F|E) = P (F) provided P (E) ≠ 0
and P(E|F) = P (E) provided P (F) ≠ 0

Definition 3: Let E and F be two events associated with the same random experiment, then E and F are said to be independent if
P(E ∩ F) = P(E) . P (F)

Example: A die is thrown. If E is the event ‘the number appearing is a multiple of 3’ and F be the event ‘the number appearing is even’ then find whether E and F are independent ?

Example: An unbiased die is thrown twice. Let the event A be ‘odd number on the first throw’ and B the event ‘odd number on the second throw’. Check the independence of the events A and B.

Example: Three coins are tossed simultaneously. Consider the event E ‘three heads or three tails’, F ‘at least two heads’ and G ‘at most two heads’. Of the pairs (E,F), (E,G) and (F,G), which are independent? which are dependent?

Example: Prove that if E and F are independent events, then so are the events E and F′.

Example: If A and B are two independent events, then the probability of occurrence of at least one of A and B is given by 1– P(A′) P(B′)

Q) Given that the events A and B are such that P(A) = 1/2 , P(A ∪ B) = 3/5 and P(B) = p. Find p if they are (i) mutually exclusive (ii) independent.

Q) Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
(i) both balls are red.
(ii) first ball is black and second is red.
(iii) one of them is black and other is red.

Q) In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random.
(a) Find the probability that she reads neither Hindi nor English newspapers.
(b) If she reads Hindi newspaper, find the probability that she reads English newspaper.
(c) If she reads English newspaper, find the probability that she reads Hindi newspaper.

Q) The probability of obtaining an even prime number on each die, when a pair of dice is rolled is
(A) 0 (B) 1/3 (C) 1/12 (D) 1/36

Q) Two events A and B will be independent, if
(A) A and B are mutually exclusive
(B) P(A′B′) = [1 – P(A)] [1 – P(B)]
(C) P(A) = P(B)
(D) P(A) + P(B) = 1