### 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 E_{1}, E_{2},..., E_{n} are mutually exclusive and exhaustive if E_{1} ∪ E_{2} ∪ ...∪ E_{n} = S and E_{i} ∩ E_{j} = φ ∀ i ≠ j

**Equally likely outcomes:** All outcomes with equal probability.

### Probability

Number P(ω_{i}) associated with sample point ω_{i} such that

(i) 0 ≤ P(ω_{i}) ≤ 1

(ii) ∑P(ω_{i}) for all ω_{i} ∈ S = 1

(iii) P(A) = ∑P(ω_{i}) for all ω_{i} ∈ A.

The number P(ω_{i}) 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)