AlgebraHigh Yield (2-3 Qs in JEE)

Probability & Distributions Formulas

Conditional probability, Bayes theorem, independent events, total probability, and binomial distribution.

7 Formulas•1,001+ Questions in Bank

Core Formulas & Mathematical Identities

P(A|B) = \frac{P(A \cap B)}{P(B)} \quad (P(B) > 0)

Probability of event A occurring given that B has already occurred.

Examiner Tip: If events are independent, P(A|B) = P(A).

P(E_i|A) = \frac{P(E_i) P(A|E_i)}{\sum_{j=1}^n P(E_j) P(A|E_j)}

Calculates reverse (posterior) probability using total probability denominator.

Examiner Tip: Write partition events E₁, E₂... clearly before evaluating branch likelihoods.

P(X = r) = {}^nC_r p^r q^{n-r}, \quad \text{Mean } \mu = np, \quad \text{Variance } \sigma^2 = npq

Probability distribution for r successes in n independent Bernoulli trials (p + q = 1).

Examiner Tip: Variance is ALWAYS strictly less than Mean for Binomial distributions (npq < np since q < 1).

P(A) = \sum_{i=1}^n P(E_i) \cdot P(A|E_i)

Law of total probability over a partition of sample space.

Examiner Tip: Forms the denominator of Bayes theorem.

P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(AB) - P(BC) - P(CA) + P(ABC)

Probability of occurrence of at least one of three arbitrary events.

Examiner Tip: Probability of exactly two events occurring = P(AB) + P(BC) + P(CA) - 3 P(ABC).

\text{Odds in favor of } A = \frac{P(A)}{P(A')} = \frac{m}{n} \implies P(A) = \frac{m}{m + n}, \quad \text{Odds against } = \frac{n}{m}

Ratio of favorable outcomes to unfavorable outcomes.

Examiner Tip: If odds in favor of an event are 3:5, its probability is 3/(3 + 5) = 3/8.

E(X) = \sum x_i p_i, \quad \text{Var}(X) = E(X^2) - [E(X)]^2 = \sum x_i^2 p_i - \mu^2, \quad \text{Var}(aX + b) = a^2 \text{Var}(X)

Mathematical expectation (weighted mean) and spread of a discrete probability distribution.

Examiner Tip: E(aX + b) = a E(X) + b. For independent random variables X and Y: Var(X ± Y) = Var(X) + Var(Y).

Frequently tested misconceptions in Probability & Distributions

•Confusing mutually exclusive events (P(A ∩ B) = 0) with independent events (P(A ∩ B) = P(A)P(B)).
•Forgetting to verify that probabilities sum to 1.