Revision: Probability Mathematics ISC (Science) ISC Class 12 CISCE

Probability measures the degree of certainty of the occurrence of an event.

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

If the range of ( X ) is an interval (a, b), then ( X ) is called a continuous random variable.

Let ( S ) be the sample space associated with a given random experiment. Then, a real-valued function ( X ) which assigns an outcome \[w\in S\] to a unique real number X(w) is called a random variable.

If the range of the real function \[X:U\to R\]is a finite set or an infinite set of real numbers, then X is called a discrete random variable.

\[P(A | B) = \frac{P(A \cap B)}{P(B)}\], 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

For two events:

• \[P(E \cap F) = P(F) \cdot P(E | F)\],\[ P(F) \neq 0. \]

• \[P(E \cap F) = P(E) \cdot P(F | E)\],\[ P(E) \neq 0. \]

For three events:

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,

Playing Cards – Key Facts

• Total cards = 52

• Red cards = 26 (Hearts, Diamonds)

• Black cards = 26 (Clubs, Spades)

• Each suit has 13 cards

• Face cards = King, Queen, Jack (Total = 12)

• 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.

• Multiplication theorem is used to find the probability of simultaneous occurrence of events.

• For two events: \[P(E \cap F) = P(E) \cdot P(F | E)\]

• Another equivalent form is \[P(E \cap F) = P(F) \cdot P(E | F)\].

• For three events: \[P(E \cap F \cap G) = P(E) \cdot P(F | E) \cdot P(G | E \cap F)\].

• Most “without replacement” questions are solved using this theorem.

• Always define events before solving a probability problem.

• 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).