Probability

🎯 Key Points

• P(A∪B)=P(A)+P(B)-P(A∩B) always; for MUTUALLY EXCLUSIVE events only, P(A∩B)=0 so it simplifies to P(A)+P(B)
• Independent events: P(A∩B)=P(A)×P(B) - this is a SPECIAL case, not the general multiplication law, which is P(A∩B)=P(A)×P(B|A)
• Bayes' theorem reverses a conditional: given P(A|B), find P(B|A) - essential whenever a problem gives "probability of evidence given hypothesis" but asks for "probability of hypothesis given evidence"
• Binomial: mean=np, variance=npq (always less than mean since q<1); Poisson: mean=variance=λ - the EQUAL mean/variance is the signature that distinguishes Poisson from binomial

Probability

Probability measures the likelihood of events occurring, ranging from 0 (impossible) to 1 (certain).

Basic Definitions

• Sample space (S): Set of all possible outcomes
• Event (E): Subset of sample space
• P(E) = n(E)/n(S) (classical definition)
• 0 ≤ P(E) ≤ 1; P(S) = 1; P(∅) = 0

Addition Law

• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Mutually exclusive events: P(A ∪ B) = P(A) + P(B) [since P(A ∩ B) = 0]
• Complement: P(A') = 1 - P(A)

Conditional Probability

• P(A|B) = P(A ∩ B) / P(B)
• Independent events: P(A ∩ B) = P(A) × P(B); P(A|B) = P(A)
• Multiplication law: P(A ∩ B) = P(A) × P(B|A)

Bayes' Theorem

P(B|A) = P(A|B) × P(B) / P(A): used to update probability given new evidence.

Probability Distributions

• Binomial: P(X=r) = nCr × pʳ × qⁿ⁻ʳ (n trials, p success, q=1-p). Mean = np, Variance = npq
• Poisson: P(X=r) = e^-λ × λʳ/r! (rare events). Mean = Variance = λ
• Normal: Continuous, bell-shaped. Standardize: Z = (X-μ)/σ

Common Experiments

• Dice (6 outcomes): P(even) = 3/6 = 1/2
• Coin (2 outcomes): P(head) = 1/2
• Cards (52): 4 suits × 13 ranks; P(ace) = 4/52 = 1/13

🚀 JEE Advanced Edge

Total probability theorem as the engine behind Bayes' theorem: If B₁,B₂,...,Bₙ partition the sample space (mutually exclusive, exhaustive), then P(A) = ΣP(Bᵢ)·P(A|Bᵢ) - this "law of total probability" is what computes the denominator P(A) inside Bayes' theorem when it isn't given directly, making the two theorems a matched pair rather than independent tools.

Why "at least one" probability problems are solved via the complement: Computing P(at least one success in n trials) directly requires summing P(exactly 1)+P(exactly 2)+...+P(exactly n) - tedious. Instead, P(at least one) = 1 - P(none), and P(none) is a single easy term to compute, making the complement approach dramatically faster whenever a problem contains the phrase "at least one."

Worked problem: Three machines A, B, C produce 25%, 35%, 40% of a factory's output respectively, with defect rates 5%, 4%, 2%. A randomly selected item is found defective. Find the probability it came from machine A. Approach: By total probability, P(defective) = 0.25×0.05 + 0.35×0.04 + 0.40×0.02 = 0.0125+0.014+0.008 = 0.0345. By Bayes: P(A|defective) = P(A)×P(defective|A)/P(defective) = 0.0125/0.0345 ≈ 0.362 (about 36.2%).

Worked Example: Conditional Probability

A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. Find the probability that both are red.

P(1st red) = 5/8. Given 1st is red, only 4 red remain out of 7 balls: P(2nd red | 1st red) = 4/7.

P(both red) = (5/8) × (4/7) = 20/56 = 5/14. Without replacement: the denominator decreases by 1 for each draw, and the number of favourable outcomes also changes.

Worked Example: Addition Rule (Mutually Non-exclusive Events)

A card is drawn from a deck of 52. Find P(red or face card).

P(red) = 26/52, P(face card) = 12/52, P(red AND face card) = 6/52 (6 red face cards). By addition rule: P(red ∪ face) = 26/52 + 12/52 − 6/52 = 32/52 = 8/13. Subtract the intersection to avoid double-counting.

Multiplication Theorem and Independent Events

• Multiplication theorem: P(A ∩ B) = P(A)·P(B|A) = P(B)·P(A|B), valid whenever P(A) and P(B) are non-zero.
• For three events: P(A ∩ B ∩ C) = P(A)·P(B|A)·P(C | A ∩ B).
• Independent events: A and B are independent if P(A ∩ B) = P(A)·P(B); equivalently P(A|B) = P(A) - one event's occurrence does not change the other's probability.
• Independent is NOT the same as mutually exclusive: two events with non-zero probability cannot be both (mutually exclusive means P(A ∩ B) = 0).
• If A and B are independent, so are A' and B, A and B', and A' and B'.

Total Probability Theorem

If E₁, E₂, …, Eₙ form a partition of the sample space (mutually exclusive, exhaustive, each with non-zero probability) and A is any event, then:

P(A) = P(E₁)·P(A|E₁) + P(E₂)·P(A|E₂) + … + P(Eₙ)·P(A|Eₙ)

This computes the overall probability of A by conditioning on which case E_i occurred, and supplies the denominator P(A) used inside Bayes' theorem.

Random Variable, Mean and Variance

• A random variable X is a real-valued function on the sample space; a probability distribution lists each value xᵢ with its probability pᵢ, where every pᵢ ≥ 0 and Σpᵢ = 1.
• Mean (expectation): μ = E(X) = Σ xᵢ pᵢ.
• Variance: Var(X) = Σ (xᵢ − μ)² pᵢ = E(X²) − [E(X)]², where E(X²) = Σ xᵢ² pᵢ.
• Standard deviation: σ = √Var(X) (measures spread about the mean).

Bernoulli Trials and Binomial Distribution

• Bernoulli trials: independent, repeated trials with exactly two outcomes (success/failure) and the same success probability p on every trial.
• Binomial distribution: for n trials, P(X = r) = ⁿC_r p^r q^n−r, with q = 1 − p and r = 0, 1, …, n.
• Mean = np, Variance = npq; the variance is always less than the mean because q < 1.
• Written as B(n, p); the terms are the successive terms of the expansion of (q + p)ⁿ.

Conditional Probability: Definition and Properties

Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred.

• It is defined as P(A|B) = P(A intersection B) divided by P(B), provided P(B) is greater than 0.
• P(A|B) lies between 0 and 1, and P(S|B) = 1 where S is the whole sample space.
• If A and B are mutually exclusive events (they cannot happen together), then P(A|B) = 0.
• Conditioning on B effectively shrinks the sample space to the outcomes contained in B.

Independent Events versus Mutually Exclusive Events

These two ideas are often confused, but they describe very different relationships between events.

• Events A and B are independent when the occurrence of one does not affect the probability of the other, i.e. P(A intersection B) = P(A) times P(B).
• For independent events, P(A|B) = P(A) and P(B|A) = P(B).
• Events are mutually exclusive when they cannot occur together, so P(A intersection B) = 0.
• Two events with non-zero probabilities cannot be both independent and mutually exclusive at the same time.

Bayes' Theorem

Bayes' theorem reverses conditional probabilities, updating the chance of a cause given an observed effect.

• If E₁, E₂, ..., En are mutually exclusive and exhaustive events (a partition) with an event A, then P(Ei|A) = [P(Ei) times P(A|Ei)] divided by the sum over all j of [P(Ej) times P(A|Ej)].
• The prior probabilities P(Ei) are the initial chances of the causes before A is observed.
• The posterior probability P(Ei|A) is the revised chance of cause Ei after A is known to have occurred.
• The denominator is exactly the total probability of A, computed by the total probability theorem.

Probability Distribution of a Random Variable

A random variable X assigns a real number to each outcome of a random experiment.

• A probability distribution lists all possible values of X together with their probabilities P(X = xi).
• Every probability must satisfy 0 less than or equal to P(X = xi) less than or equal to 1.
• The sum of all the probabilities in the distribution equals 1.
• A discrete random variable takes isolated (countable) values, such as the number of heads in three tosses of a coin.