Question

The probability of an event A lies between 0 and 1, both inclusive. Which mathematical expression best describes this statement?
A. \[0\le P\left( A \right)\le 1\]
B. $0=P\left( A \right)\le 1$
C. $0\le P\left( A \right)=1$
D. $0\le P\left( A \right)\ge 1$

Answer 1

Hint: We must look at the basic definition of mutually inclusive and mutually exclusive events in probability to arrive at a conclusion regarding the given statement. We also know that the probability of any event is given as $=\dfrac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$.

Complete step-by-step answer:
It is given in the question that the probability of an event A lies between 0 and 1 and that both are inclusive, and we must find the mathematical expression that best describes this statement. To reach any conclusion, we have to look at the basic definition of mutually exclusive events and mutually inclusive events.
If we suppose two events as A and B, so when these two events A and B cannot occur at the same time, they are called mutually exclusive events. Mutually exclusive events have no common output. They are represented diagrammatically as follows.

For example, let us take a set of prime numbers and composite numbers. Let A be the set of prime numbers and B be the set of composite numbers. Both are mutually exclusive events as there is no number that is both prime and composite.
Whereas mutually inclusive events share some events or they have some events that are common to each other. It is represented diagrammatically as follows.

For example, if we have a set of 5 natural numbers, $S=\left\{ 1,2,3,4,5 \right\}$. Let A be the set of prime numbers $A=\left\{ 2,3,5 \right\}$ and B be the set of numbers less than 4, $B=\left\{ 1,2,3 \right\}$. Then we have 2,3 that are common in set A and B. It means that 2 and 3 are prime numbers as well as less than 4, so this event is called as mutually inclusive events.
So, from the above discussion, we can choose the best option for the probability of an event A that lies between 0 and 1 as \[0\le P\left( A \right)\le 1\].

Therefore, option A, \[0\le P\left( A \right)\le 1\] is the correct answer.

Note: We have already discussed mutually inclusive and exclusive events, so to make it clearer, we must know the definition of dependent and independent events too. Dependent events are defined as 2 or more events whose outcomes affect each other or the outcome of the two events depend on each other. Whereas independent events are defined as 2 or more events whose outcomes do not affect each other.