What is the relation between the quantity of information of $A_i$ and $A$, where $A=\bigcup_iA_i$?
I believe that the correct formula is
$$
I(A) = \log \frac{1}{\Pr[A]}.
$$

In this case, we have
$$
I(A_i) = \log \frac{4}{\Pr[A]} = \log \frac{1}{\Pr[A]} + \log 4 = I(A) + \log 4.
$$