Mathematics Mutually Exclusive & Exhaustive Event

Equally Likely Events

Events are said to be equally likely when no particular event preference to occur in relation to the other
event.
Example:
(i) The outcomes as result throwing a die are equally likely, as no particular face is more likely to occur as
compared to other faces. That is why we normally write as fair die or unbiased die.
(ii) The outcomes as result of drawing a card from a well shuffled pack of 52 playing cards are equally likely
to occur. Each card is as likely to be withdrawn as any other card.
(iii) How ever getting of a total of 7 is not as equally likely as getting of a total of 12 when a pair of dice are
rolled once. It is also to be noted that it is 6 times more likely to get a total of 7 than to get a total of 12
in a single throw with the pair of dice.

Mutually Exclusive / Disjoint / Incompatible Events :

Two events A and Bare said to be mutually exclusive events if their simultaneous occurrence is impossible,
i.e. both the events can not occur together.

`P(A cap B) = phi`

Example:
(i) In throwing a fair die, to events A and Bare such that
A: getting an odd number
B :getting an even number
then A & Bare mutually exclusive events.
(ii) In drawing a card from a well shuftled pack of 52 playing card two events A and Bare such that
A : getting an ace
B : getting a red card
then A and Bare not mutually exclusive events.
(iii) For three events "A", "B" and "C" which are mutually exclusive and exhaustive, the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is a certainty and is given by the sum of the probabilities of the individual events.
`P(A ∪ B ∪ C) = P(A) + P(B) + P(C)` [Mutually Exclusive]
`P(A ∪ B ∪ C) = P(A) + P(B) = 1` [Mutually Exclusive and Exhaustive]

Exhaustive Events

If `E_1, E_2, .... , E_n` are n events associated with an experiment whose sample space is S and if `E_1 cup E_2 cup E_3 cup ...... cup E_n = underset (i= 1) (overset (n) cupE_i) = S`
then `E_1, E_2, ...... , E_n` are called exhaustive events. In other words, events `E_1, E_2, ...... , E_n`, are said to be
exhaustive if atleast one of them necessarily occurs whenever the experiment is performed.

Further, if `E_i E_j = phi` for `i nej` i.e. , events `E_i` , and `E_j` are pairwise disjoint and ,`underset (i= 1) (overset (n) cupE_i)` then events `E_1, E_2, .... , E_n` are called mutually exclusive and exhaustive events.

e.g.
For three events "A", "B" and "C" which are mutually exclusive and exhaustive, the probability that atleast one of the events would occur i.e. the probability of the occurrence of the union of the events is a certainty and is given by the sum of the probabilities of the individual events.

`P(A ∪ B ∪ C) = 1` [Exhaustive]
`P(A ∪ B ∪ C) = P(A) + P(B) = 1` [Mutually Exclusive and Exhaustive]