Why probability of an event always lie between 0 and 1?

Probability refers to the extent of occurrence of events. When an event occurs like throwing a ball, picking a card from the deck, etc, then there must be some probability associated with that event.

Mutually Exclusive Event:
Given two events A and B, if both of these events have nothing in common i.e.  A ∩ B = ∅ then, the probability of the intersection of these events will also be equal to zero i.e. P(A ∩ B) = 0. Such events are known as Mutually Exclusive Events.

Sample Space:
It is a set of all the possible outcomes of an experiment. In this article, we will denote a sample space by ‘S’.

Now, there are three important axioms related to Probability, which will really help us in proving the above statement. So, let’s have a look at these axioms-

1. Probability of an event will always be greater than or equal to zero i.e. P(A) >= 0 for any event A.
2. Probability of a Sample Space will always be equal to 1 i.e. P(S) = 1
3. Given some mutually exclusive events, the probability of the union of all these mutually exclusive events will always be equal to the summation of the probability of individual events i.e. P(A₁ ∪ A₂ ∪ A₃ ∪ A₄ … ∪ A_N) = P(A₁) + P(A₂) + P(A₃) +P (A₄) + …. + P(A_N)

Problem Statement:
The task here is to prove that the probability of A will always lie between 0 and 1 i.e. 0 <= P(A) <= 1.

Solution: Consider event A. Below are the steps for the proof of the above problem statement-

• According to axiom 1, the probability of an event will always be greater than or equal to 0.

P(A) >= 0 (According to Axiom 1)  --- (1)

• The probability of a sample space will be equal to the probability of the intersection of A and (S – A) i.e.

S = A + (S - A)
P(S) = P(A + (S - A))

• Since A and (S – A) are two mutually exclusive events. So, according to axiom 3, it can be written-

P(A + (S - A)) = P(A) + P(S - A)

• This implies,  

P(S) = P(A) + P(S - A)) --- (2)

• Now, from axiom 1, it can be said that the P(S – A) will always be greater than or equal to zero i.e. P(S – A) >= 0.
• If something positive is added to a given value, its value will always increase. Since, P(S – A) >=0, it can be said that P(A) can’t be greater than P(S). Otherwise, equation (2) will not hold true. 
• This means-

P(S) >= P(A)

• From axiom 2, the probability of a Sample Space always equals 1. So, this means- 

1 >= P(A)
   or
P(A) <= 1 --- (3)

• From, equation (1) and (3), it can be shown that-

0 <= P(A) <= 1

This proves that the probability of an event will always lie between 0 and 1.