If $P(A)=P(B)=1$, then $P(A\cap B)=1$

Question

This is my reasoning:

Using the probability law that

$$P(A \cup B)=P(A)+P(B)-P(A\cap B)$$

Since $P(A)+P(B) = 1$, and we know that the total probability cannot be greater than $1$. So, I can say that $P(A\cap B)$ must be equal to $1$.

Am I correct?

Answer 1

Yup.

We have $$P(A\cup B) + P(A \cap B)=2$$

Hence, we must have $P(A \cup B)=P(A \cap B)=1$.

Answer 2

Your proof is correct. I believe it is essential to understand the meaning of two events having probability 1.

If $ P(A)= 1 $ , then the probability of any event other than A is $0$. So there cannot exist an event B such that $P(B)=1$, unless the events A and B are the same events.

Since $A$ and $B$ are the same events, then $ P(A \bigcup B)= P(A) = P(B) =1 $.

Now, applying the rule $ P( A \bigcap B) = P(A) + P(B) - P( A \bigcup B) $, then

$ P( A \bigcap B) = 1+1-1 $

Hence $ P( A \bigcap B) =1 $