Question

Which of the following is/are correct if the probability of the intersection of two events A and B is zero?
A). A and B are mutually exclusive
B). A and B has one or more common events
C). P(A$\cap$B) $= 0$
D). None of these

Answer 1

When two events don’t have any intersection then we call them mutually exclusive events. Then their probability of intersection is also zero that is, if $A$ and $B$ are any two events that are mutually exclusive then $P(A \cap B) = 0$.

Complete step-by-step solution:
It is given that there are two events A and B whose probability of intersection is zero.
Since the probability of these two events (A and B) is zero, we can write it as P(A$\cap$B)$= 0$.
We know that if any two events (say A and B) whose probability of intersection is zero then, they are known as mutually exclusive events (that is, $A$ and $B$ are mutually exclusive events).
Now let us see the options.
Option (a) $A$ and $B$ are mutually exclusive, this option is correct since we know that if the probability of the intersection of any two events is zero then they are called mutually exclusive.
Option (b) $A$ and $B$ has one or more common events, this cannot be the correct answer because when the probability of the intersection of any two events is zero then there won’t be any events that occur commonly. Thus, it cannot be the right answer.
Option (c) $P(A \cap B) = 0$, is the correct answer since we already know that if the probability of the intersection of any two events is zero then, it can be written as $P(A \cap B) = 0$.
Option (d) None of these, this option cannot be the right answer since we got two of the options as a correct answer.
Thus, the right answers for this problem are an option (a) and (c).

Note: Mutually exclusive events are also called disjoint events since there is no common event occurring. For instance, an event of occurring head and event of occurring tail on tossing a coin are mutually exclusive events. The events are individually known as independent events.