CBSE · Class 12 · Mathematics · Chapter 13

Probability — Formula Sheet

Board Formulas

12 formulas

1.Conditional Probability★
$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}, P (B) > 0$
Probability of A given B has happened. Restrict the sample space to B.
2.Multiplication Rule★
$P (A \cap B) = P (B) \cdot P (A ∣ B) = P (A) \cdot P (B ∣ A)$
Chain of two events. Useful when one probability is easy to compute and the conditional is given.
3.Independence
$P (A \cap B) = P (A) \cdot P (B) \Leftrightarrow A, B independent$
Equivalent forms: P(A|B) = P(A), P(B|A) = P(B). Very different from mutually exclusive.
4.Total Probability Theorem
$P (B) = \sum_{i} P (A_{i}) P (B ∣ A_{i})$
For a partition A₁, A₂, ..., Aₙ of the sample space, sum weighted conditional probabilities.
5.Bayes' Theorem★
$P (A_{i} ∣ B) = \frac{P ( A _{i} ) P ( B ∣ A _{i} )}{j \sum P ( A _{j} ) P ( B ∣ A _{j} )}$
Reverses the direction of conditioning. Numerator: prior × likelihood; denominator: total probability of B.
6.Complement Rule
$P (A^{c}) = 1 - P (A), P (at least one) = 1 - P (none)$
'At least one' problems ⇒ complement first. Almost always shorter than direct counting.
7.Random Variable Expectation★
$E (X) = \sum_{i} x_{i} P (X = x_{i})$
Weighted average of possible values, weights = probabilities.
8.Variance
$Var (X) = E (X^{2}) - [E (X)]^{2}$
Standard shortcut: compute E(X²) and E(X) separately. Always non-negative.
9.Binomial Distribution★
$P (X = r) = (r n) p^{r} q^{n - r}, q = 1 - p, r = 0, 1, \dots, n$
Fixed n trials, independent, each with success probability p. X counts successes.
10.Mean of Binomial
$μ = E (X) = n p$
Intuitive: expected number of successes in n independent trials.
11.Variance of Binomial
$Var (X) = n pq, σ = n pq$
Maximum when p = ½ (highest uncertainty per trial).
12.Bernoulli Trial
$P (X = 1) = p, P (X = 0) = 1 - p$
A single yes/no experiment. Binomial with n = 1. Basis for the binomial distribution.