Class 12 Maths Notes: Probability – Mutually Exclusive, Exhaustive and Independent Events

A set of events is said to be mutually exclusive if the occurrence of one of them precludes the occurrence of any of the other events. For instance, when a pair of dice is tossed, the events a sum of 4 occurs’, ‘a sum of 10 occurs’ and ‘a sum of 12 occurs’ are mutually exclusive. Simply speaking, if two events are mutually exclusive they can not occur simultaneously. Using set theoretic notation, if A₁, A₂,.. A_n be the set of mutually events then A_i ∩ A_j = Ф for i ≠ j and 1 ≤ i, j≤ n.

Independent Events

Events are said to be independent if the occurrence or non-occurrence of one does not affect the occurrence or non-occurrence of other. For instance, when a pair of dice is tossed, the events ‘first die shows an even number’ and ‘second die shows an odd number’ are independent. As the outcome of second die does not affect the outcome of first die. It should also be noted that these two events are not mutually exclusive as they can occur together.

Remarks

• Distinction between mutually exclusive and independent events should be clearly made. To be precise, concept of mutually exclusive events is set theoretic in nature while the concept of independent events is probabilistic in nature.
• If two events A and B are mutually exclusive, they would be strongly dependent as the occurrence of one precludes the occurrence of the other.

Exhaustive Event

A set of events is said to be exhaustive if the performance of random experiment always result in the occurrence of atleast one of them. For instance, consider a ordinary pack of cards then the events ‘drawn card is heart’, drawn card is diamond’, ‘drawn card is club’ and ‘drawn card is spade’ is set of exhaustive event. In other words all sample points put together (i.e. sample space itself) would give us an exhaustive event.

If ‘E’ be an exhaustive event then P(E) = 1.

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