Maharashtra State Board · Class 12 · Mathematics · Part II Chapter 7

Probability Distributions — Formula Sheet

Board Formulas

12 formulas

1.Classical Definition
$P (A) = \frac{n ( A )}{n ( S )}$
Ratio of favourable outcomes to total outcomes (equally likely sample space).
2.Addition Theorem★
$P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Subtract intersection to avoid double counting. For mutually exclusive events, P(A ∩ B) = 0.
3.Conditional Probability★
$P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}, P (B) > 0$
Probability of A given B has occurred. Restricts sample space to B.
4.Multiplication Theorem
$P (A \cap B) = P (A) \cdot P (B ∣ A) = P (B) \cdot P (A ∣ B)$
For independent events P(A ∩ B) = P(A)·P(B).
5.Total Probability Theorem★
$P (B) = \sum_{i} P (A_{i}) \cdot P (B ∣ A_{i})$
A_i are a partition of the sample space (mutually exclusive and exhaustive).
6.Bayes Theorem★
$P (A_{i} ∣ B) = \frac{P ( A _{i} ) \cdot P ( B ∣ A _{i} )}{\sum _{j} P ( A _{j} ) \cdot P ( B ∣ A _{j} )}$
Reverse conditional probability using prior P(A_i) and likelihood P(B|A_i).
7.Probability Distribution of Random Variable★
$\sum_{i} p_{i} = 1, p_{i} = P (X = x_{i}) \geq 0$
Tabulate all values x_i and their probabilities p_i. Sum of column = 1.
8.Expectation (mean)★
$E (X) = μ = \sum_{i} x_{i} p_{i}$
Weighted average of outcomes weighted by their probabilities.
9.Variance and Standard Deviation
$Var (X) = \sum_{i} x_{i}^{2} p_{i} - μ^{2}; σ = Var (X)$
Expected squared deviation from the mean. Var(X) = E(X²) − [E(X)]².
10.Binomial Distribution (PMF)★
$P (X = r) = (r n) p^{r} q^{n - r}, q = 1 - p$
n independent Bernoulli trials with success probability p. Random variable X = number of successes.
11.Binomial Mean and Variance
$E (X) = n p, Var (X) = n pq$
Mean and variance of binomial. Standard deviation = √(npq).
12.Independence
$P (A \cap B) = P (A) \cdot P (B) ⟺ A, B independent$
Independence means one event does not affect the other's probability.