, }, A = { , , } and B = { , , } Obviously ‘A or B’ = A ∪ B = { , , , } ‘A and B’ = A ∩ B = { , } (iii) ‘A but not B’ = A – B = { } (iv) ‘not A’ = A ′ = { , , } PROBABILITY . . Mutually exclusive events In the experiment of rolling a die, a sample space is S = { , , , , , }. Consider events, A ‘an odd number appears’ and B ‘an even number appears’ Clearly the event A excludes the event B and vice versa.
In other words, there is no outcome which ensures the occurrence of events A and B simultaneously. Here A = { , , } and B = { , , } Clearly A ∩ B = φ, i.e., A and B are disjoint sets. In general, two events A and B are called mutually exclusive events if the occurrence of any one of them excludes the occurrence of the other event, i.e., if they can not occur simultaneously. In this case the sets A and B are disjoint.
Again in the experiment of rolling a die, consider the events A ‘an odd number appears’ and event B ‘a number less than appears’ Obviously A = { , , } and B = { , , } Now ∈ A as well as ∈ B Therefore, A and B are not mutually exclusive events. Remark Simple events of a sample space are always mutually exclusive. . .
Exhaustive events Consider the experiment of throwing a die. We have S = { , , , , , }. Let us define the following events A: ‘a number less than appears’, B: ‘a number greater than but less than appears’ and C: ‘a number greater than appears’. Then A = { , , }, B = { , } and C = { , }.
We observe that A ∪ B ∪ C = { , , } ∪ { , } ∪ { , } = S. Such events A, B and C are called