## CBSE Class 11 Maths Notes Chapter 16 Probability

Random Experiment

An experiment whose outcomes cannot be predicted or determined in advance is called a random experiment.

Outcome

A possible result of a random experiment is called its outcome.

Sample Space

A sample space is the set of all possible outcomes of an experiment.

Events

An event is a subset of a sample space associated with a random experiment.

Types of Events

Impossible and sure events: The empty set Φ and the sample space S describes events. Intact Φ is called the impossible event and S i.e. whole sample space is called sure event.

Simple or elementary event: Each outcome of a random experiment is called an elementary event.

Compound events: If an event has more than one outcome is called compound events.

Complementary events: Given an event A, the complement of A is the event consisting of all sample space outcomes that do not correspond to the occurrence of A.

Mutually Exclusive Events

Two events A and B of a sample space S are mutually exclusive if the occurrence of any one of them excludes the occurrence of the other event. Hence, the two events A and B cannot occur simultaneously and thus P(A ∩ B) = 0.

Exhaustive Events

If E1, E2,…….., En are n events of a sample space S and if E1 ∪ E2 ∪ E3 ∪………. ∪ En = S, then E1, E2,……… E3 are called exhaustive events.

Mutually Exclusive and Exhaustive Events

If E1, E2,…… En are n events of a sample space S and if

Ei ∩ Ej = Φ for every i ≠ j i.e. Ei and Ej are pairwise disjoint and E1 ∪ E2 ∪ E3 ∪………. ∪ En = S, then the events

E1, E2,………, En are called mutually exclusive and exhaustive events.

Probability Function

Let S = (w1, w2,…… wn) be the sample space associated with a random experiment. Then, a function p which assigns every event A ⊂ S to a unique non-negative real number P(A) is called the probability function.

It follows the axioms hold

- 0 ≤ P(wi) ≤ 1 for each Wi ∈ S
- P(S) = 1 i.e. P(w1) + P(w2) + P(w3) + … + P(wn) = 1
- P(A) = ΣP(wi) for any event A containing elementary event wi.

Probability of an Event

If there are n elementary events associated with a random experiment and m of them are favorable to an event A, then the probability of occurrence of A is defined as

The odd in favour of occurrence of the event A are defined by m : (n – m).

The odd against the occurrence of A are defined by n – m : m.

The probability of non-occurrence of A is given by P(

Addition Rule of Probabilities

If A and B are two events associated with a random experiment, then

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

Similarly, for three events A, B, and C, we have

P(A ∪ B ∪ C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)

Note: If A andB are mutually exclusive events, then

P(A ∪ B) = P(A) + P(B)