If 𝐴 and 𝐵 are two mutually
exclusive events from a sample space of a random experiment, find the probability of
𝐴 complement union 𝐵 complement.
To answer this question, we need to
recall one of De Morgan’s laws: 𝐴 complement union 𝐵 complement is equivalent to
the complement of 𝐴 intersect 𝐵. And so, it follows that the
probability of 𝐴 complement union 𝐵 complement will be the probability of the
complement of 𝐴 intersect 𝐵. Now, this law holds regardless. But these two events in this
question are mutually exclusive. And we know that for mutually
exclusive events, their intersection is the empty set. This means that the complement of
their intersection is the entire sample space, which has a probability of one.
Or more formally, we can say that
the probability of 𝐴 intersect 𝐵 complement is one minus the probability of 𝐴
intersect 𝐵. That’s one minus zero, which is
one. So, by recalling De Morgan’s law
and that for mutually exclusive events the probability of their intersection of
zero, we’ve found that for the two mutually exclusive events 𝐴 and 𝐵, the
probability of 𝐴 complement union 𝐵 complement is one.