exhaustive events. In general, if E , E , ..., E n are n events of a sample space S and if E E E E E S ... ∪ ∪ ∪ ∪ = ∪ then E , E , ...., E n are called exhaustive events .In other words, events E , E , ..., 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 = φ for i ≠ j i.e., events E i and E j are pairwise disjoint and S E ∪ , then events E , E , ..., E n are called mutually exclusive and exhaustive events .
MATHEMATICS We now consider some examples. Example Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment A: ‘the sum is even’. B: ‘the sum is a multiple of ’.
C: ‘the sum is less than ’. D: ‘the sum is greater than ’. Which pairs of these events are mutually exclusive? Solution There are elements in the sample space S = {( x , y ): x , y = , , , , , }.
Then A = {( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )} B = {( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )} C = {( , ), ( , ), ( , )} and D = {( , )} We find that A ∩ B = {( , ), ( , ), ( , ), ( , ), ( , ), ( , )} ≠ φ Therefore, A and B are not mutually exclusive events. Similarly A ∩ C ≠ φ , A ∩ D ≠ φ , B ∩ C