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

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

Vedantu Content Team · Accepted Answer

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 SNow, 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 setAlso 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.