Question

Assume that P (A) =P (B). Show that A=B.

Answer 1

Hint: Here we will consider the events and use the properties of probability to prove A = B.

Complete step-by-step answer:

Now we have been given that $P(A) = P(B)$
We need to show that $A = B$
Now let us suppose we have a sample event x such that $x \in A$
Now clearly $A \in P(A)$and $P(A) = P(B)$
Hence we can say that our $x \in C$ for some $C \in P(B)$
Now C is a sample event of B so obviously $C \subset B$.
Hence in other words we can say that $x \in B$.
But as mentioned above x is a sample event of A so $A \subset B \to (1)$.
Now let y be any sample event such that $y \in B$.
Now $B \in P(B)$and $P(B) = P(A)$.
So we can say that $y \in D$ for some $D \in P(A)$
Now $D \subset A$. So our $y \in A$. But as stated above that y is a sample venet of B
Hence $B \subset A \to (2)$
Clearly from eq 1 and 2 we can say that $A = B$.
Hence Proved.

Note: While solving such problems always start by considering a sample element of an event of given probability and then proceed to find the proof.