Class 12 Maths - Probability

1. Conditional Probability

$P(A|B) = \frac{P(A \cap B)}{P(B)}$ (Probability of A given B has occurred).

2. Independent Events

Two events A and B are independent if: $P(A \cap B) = P(A) \cdot P(B)$

3. Bayes' Theorem (The Most Important!)

Used to find reverse probability. $$P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum P(E_j)P(A|E_j)}$$

4. Probability Distribution

Mean (Expectation) of variable X: $\mu = E(X) = \sum x_i p_i$

Variance: $\sigma^2 = E(X^2) - [E(X)]^2$

5. Bernoulli Trials

Trials are independent and have only two outcomes: Success (p) and Failure (q). $P(X=r) = {}^nC_r p^r q^{n-r}$

Bayes' Theorem is a guaranteed 5-mark question! Practice the 'Factory Bolt' or 'Doctor Diagnosis' problems. 🎲📊