An experiment whose outcomes cannot be predicted or determined in advance is called a random experiment.
A possible result of a random experiment is called its outcome.
A sample space is the set of all possible outcomes of an experiment.
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.
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.
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() = 1 – P(A).
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)