I have a set of binary strings, length $8$ bits, so I have that the probability that these strings have two bits equal to $11$ in the least significant positions is $1/4$ (event $A_1$)
If I want to know the probability that these strings have $2$ bits equal to $11$ in the most significant position is $1/4$ also (event $A_2$)
I would like to know why it appears in the book that I am reading that:
$$\Pr[A_1 \cup A_2] < \Pr[A_1] + \Pr[A_2]$$
from what I got the union of those probabilities is $1/2$ which is equal to the sum of their individual probabilities.
Am I missing something?