Download Class 12 Mathematics (CBSE) – Probability | Complete Chapter Notes & Solved Examples and more Schemes and Mind Maps Mathematics in PDF only on Docsity!
Chapter 1: Probability Foundations
1.1 Introduction
Probability theory is a branch of mathematics concerned with the analysis of random phenomena. Its foundation lies in the study of sample spaces, events, and the assignment of probabilities according to certain axioms.
1.2 Sample Space and Events
- Sample Space (S): The set of all possible outcomes of an experiment.
S = {all possible outcomes}
Example: For a fair die roll, S = { 1 , 2 , 3 , 4 , 5 , 6 }.
- Event: A subset of the sample space. Example: The event of rolling an even number is E = { 2 , 4 , 6 }.
1.3 Probability Axioms (Kolmogorov)
For a sample space S, the probability function P satisfies:
- Non-negativity: P (A) ≥ 0 for all A ⊆ S.
- Normalization: P (S) = 1.
- Additivity: If A and B are disjoint events, then
P (A ∪ B) = P (A) + P (B).
1.4 Classical Definition of Probability
If all outcomes in S are equally likely:
P (A) = Number of outcomes in A Number of outcomes in S
Example: Probability of getting an even number in a die roll:
P (E) =
1.5 Conditional Probability
The probability of A given B is:
P (A|B) =
P (A ∩ B)
P (B)
, if P (B) > 0.
Example: For a deck of 52 cards, let A = drawing a heart, B = drawing a face card. We can compute P (A|B) accordingly.
1.6 Independence of Events
Two events A and B are independent if:
P (A ∩ B) = P (A) · P (B).
Example: Rolling two dice: Event A = ”first die shows 3”, Event B = ”second die shows even”. They are independent.
1.7 Law of Total Probability
If {B 1 , B 2 ,... , Bn} is a partition of S:
P (A) =
X^ n
i=
P (A|Bi)P (Bi).
Formula 2: Complement Rule
P (E′) = 1 − P (E)
Example: If P (Rain) = 0.3, find P (No Rain).
P (No Rain) = 1 − 0 .3 = 0. 7
Formula 3: Mutually Exclusive Events
P (A ∪ B) = P (A) + P (B) if A ∩ B = ∅
Example: Rolling a die: A = {Even}, B = {Odd}.
P (A) =
, P (B) =
P (A ∪ B) =
Formula 4: General Addition Rule
P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
Example: A card is drawn from a deck. A = red card, B = king.
P (A) =
, P (B) =
, P (A ∩ B) =
P (A ∪ B) =
Formula 5: Conditional Probability
P (A|B) =
P (A ∩ B)
P (B)
Example: A bag has 3 red, 2 blue. If one ball is drawn, A = red, B = any ball. P (A ∩ B) =
, P (B) = 1
P (A|B) =
3 5 1
Formula 6: Independent Events
P (A ∩ B) = P (A) · P (B)
Example: Tossing a fair coin and rolling a fair die: A = head, B = 6.
P (A) =
, P (B) =
P (A ∩ B) =
Formula 7: Coin Toss (Unbiased)
For n tosses:
P (k Heads) =
n k
2 n Example: In 3 tosses, find P (2 Heads):
P =
2
Formula 8: Coin Toss (Biased)
If P (Head) = p, P (Tail) = q = 1 − p:
P (k Heads in n) =
n k
pkqn−k
Example: For p = 0.6, n = 2, k = 2:
P =
(0.6)^2 (0.4)^0 = 0. 36
Formula 9: Dice – Single Roll
P (Number) =
Example: Probability of rolling a 4:
P =
Formula 15: Expected Value for Coin Toss
For n tosses, unbiased:
E[Heads] = n ·
Example: 5 tosses:
E[Heads] = 5 ·
Probability Formulas with Examples
1. Probability of a Specific Prime Outcome
Formula: P (Specific Prime) =
3 k + 3 Example: In a bag containing k groups of 3 balls each plus 3 extra balls labeled as prime numbers, find the probability of drawing the prime number
- Let k = 4. P =
2. Probability of a Specific Non-Prime Outcome
Formula: P (Specific Non-Prime) =
3 k + 3 Example: A bag contains k groups of 3 balls each plus 3 extra non-prime balls. If k = 5, find the probability of drawing the non-prime number 4.
P =
3. Probability of Getting a Prime (Any Prime)
Formula: P (Any Prime) = 3 k 3 k + 3
k k + 1 Example: If k = 6 in a set-up where 3k items are prime and 3 are non-prime, find the probability of drawing any prime number.
P =
4. Probability of Getting a Non-Prime (Any Non-Prime)
Formula: P (Any Non-Prime) =
3 k + 3
k + 1 Example: If k = 4 in a scenario where 3 items are non-prime and 3k are prime, find the probability of getting any non-prime number.
P =
Probability MCQs – GATE / CSIR-NET Level
Easy (Concept Reinforcement)
- A bag contains 3 red and 2 blue balls. If a ball is drawn at random, let A = event the ball is red, B = event the ball is any color. Find P (A ∩ B). Options: (a) 25 (b) 35 (c) 1 (d) 12 Answer: (b)
- A die is rolled once. Let A = event of getting an even number, B = event of getting a number divisible by 3. Find P (A ∪ B). Options: (a) 13 (b) 23 (c) 46 (d) 12 Answer: (b)
- Two fair coins are tossed. Let A = event that both are heads, B = event that at least one head appears. Find P (A|B). Options: (a) (^14) (b) 12 (c) 13 (d) 23 Answer: (c)
- In a factory, 40% machines are type-I and 60% are type-II. Type-I produces 4% defectives, type-II produces 2% defectives. A product is selected at random and is defective. Find the probability it came from type-I machine. Answer: 0 .5714.
- An experiment consists of tossing a die twice. Let A = event sum is even, B = event both numbers are same. Find P (A|B) and P (B|A). Answer: P (A|B) = 12 , P (B|A) = 13.
- Three urns: U 1 has 2 red, 3 white; U 2 has 4 red, 1 white; U 3 has 3 red, 2 white. An urn is chosen at random and one ball is drawn. It is red. Find the probability it came from U 2. Answer: 49.
- Let A and B be independent with P (A) = 0.4, P (B) = 0.7. Find P (A ∪ B), P (A ∩ B), and P (A|B). Answer: P (A ∪ B) = 0.82, P (A ∩ B) = 0.28, P (A|B) = 0.4.
- A system has two components in series: C 1 works with probability 0.9, C 2 works with probability 0.8. If they work independently, find system reliability and probability that exactly one works. Answer: Reliability = 0.72, exactly one works = 0.26.
- P (A) = 0.5, P (B) = 0.4, P (A ∩ B) = 0.25. Find P (A ∩ B), P (B ∩ A), P (A ∩ B). Answer: 0 .15, 0.25, 0.35.
- A biased coin has probability p of heads. Tossed 3 times, find P (exactly two heads|first toss Answer: 2 p(1 − p).
- Suppose P (A) = 0.7, P (B|A) = 0.6, P (B|A) = 0.3. Find P (B) and P (A|B). Answer: P (B) = 0.51, P (A|B) ≈ 0 .8235.
