Revision: Introduction to Probability Theory Mathematics HSC Science Class 11 Tamil Nadu Board of Secondary Education

If A and B are two events associated with the same sample space of a random experiment, the conditional probability of the event A given that B has occurred, i.e. P(A∣B), is given by

\[P(A | B) = \frac{P(A \cap B)}{P(B)}\], where \[P(B) \neq 0\].

\[P(B | A) = \frac{P(A \cap B)}{P(A)}\], where \[P(A) \neq 0\].

Multiplication rule: \[P(A \cap B) = P(B) \cdot P(A | B) = P(A) \cdot P(B | A)\].

Complement form: \[P(A' | B) = 1 - P(A | B)\], P(B) ≠ 0

If B₁, B₂,..., B_n are mutually exclusive and exhaustive events and if A is an event consequent to these B_i's, then for each i = 1, 2, 3, ..., n,

• Conditional probability means probability under a given condition.

• The formula is \[P(A | B) = \frac{P(A \cap B)}{P(B)}\], where \[P(B) \neq 0\].

• First restrict the sample space according to the given condition, then count the favourable outcomes.

• The numerator represents outcomes common to both events.

• Do not confuse P(A | B) with P(B | A).

• For independent events, P(A | B) = P(A), provided P(B) ≠ 0.

• Bayes' Theorem works from effect to cause.

• Prior probability means “before observation”.

• Posterior probability means “after observation”.

• The observed event is usually given in the question statement.

• The required answer is often a probability of the form P(cause ∣ observed event).