Terms

Sample space: The set of all possible outcomes

Sample points: Elements of sample space

Event: A subset of the sample space

Impossible event: The empty set

Sure event: The whole sample space

Complementary event or 'not event': The set A' or S - A

Event A or B: The set A ∪ B

Event A and B: The set A ∩ B

Event A and not B: The set A - B

Mutually exclusive event: A and B are mutually exclusive if A ∩ B = φ

Exhaustive and mutually exclusive events: Events E1, E2,..., En are mutually exclusive and exhaustive if E1 ∪ E2 ∪ ...∪ En = S and Ei ∩ Ej = φ ∀ i ≠ j

Equally likely outcomes: All outcomes with equal probability.

Probability

Number P(ωi) associated with sample point ωi such that

(i) 0 ≤ P(ωi) ≤ 1

(ii) ∑P(ωi) for all ωi ∈ S = 1

(iii) P(A) = ∑P(ωi) for all ωi ∈ A.

The number P(ωi) is called probability of the outcome ω.

Probability of an event

For a finite sample space with equally likely outcomes, Probability of an event P(A) = n(A)/n(S), where n(A) = number of elements in the set A, n(S) = number of elements in the set S.

If A and B are any two events, then

P(A or B) = P(A) + P(B) – P(A and B)

P(A ∪ B) = P(A) + P(B) – P(A ∩ B) 

If A and B are mutually exclusive, then

P(A or B) = P(A) + P(B) 

If A is any event, then

P(not A) = 1 – P(A)

Conditional Probability

If E and F are two events associated with the same sample space of a random experiment, then the conditional probability of the event E under the condition that the event F has occurred, written as P (E|F), is given by

P(E|F) = P(E ∩ F)/P(F)

Properties of Conditional Probability

Let E and F be events associated with the sample space S of an experiment. Then

  • P(S|F) = P(F|F) = 1
  • P[(A∪B)|F] = P(A|F) + P(B|F) – P[(A∩B|F)], where A and B are any two events associated with S.
  • P (E′|F) = 1 – P (E|F)