Question: If E and F are events with \[P\left( E \right)\le P\left( F \right)\] and \[P\left( E\cap F \right)>...

Question

If E and F are events with $P\left( E \right)\le P\left( F \right)$ and $P\left( E\cap F \right)>0$ , then
(a) Occurrence of E $\Rightarrow$ Occurrence of F
(b) Occurrence of F $\Rightarrow$ Occurrence of E
(c) non occurrence of E $\Rightarrow$ non occurrence of F
(d) none of the above implications holds

Answer 1

Solution

Hint : In this question, from the given conditions we first need to find the possible relation between the events E and F. Then we can conclude the occurrence and nonoccurrence relation between those two events.

Complete step-by-step answer :
SAMPLE SPACE: The set of all possible outcomes of an experiment is called the sample space of the experiment denoted by S.
OCCURRENCE OF AN EVENT: An event associated to a random experiment is said to occur, if any one of the elementary events associated to it is an outcome.
Now, from the given conditions in the question we have
$P\left( E \right)\le P\left( F \right)$
$P\left( E\cap F \right)>0$
Let us consider the sample space as S
Now, from the first condition given above we have
$P\left( E \right)\le P\left( F \right)$
Now, we have that E is a subset of S, F is a subset of S in which F is the larger set
Also given the condition that $P\left( E\cap F \right)>0$ which means that the events E and F have something in common.
Now, from the above two conditions we can say that E and F are dependent or independent of each other but have something in common.
Now, we have the possibility that E lies completely inside For some part of E may be present in F.
Now, it gives the conclusion that the occurrence and non occurrence of these events does not completely depend on each other.
Hence, the correct option is (d).

Note : It is important to note that as the probability of F is greater than E it implies that F is a larger set than E. This helps in concluding about the relation between those 2 events.It is also to be noted that as they are not mutually exclusive events they can be independent events which helps in commenting about the occurrence and nonoccurrence.