I am trying to solve a problem. Two dice (1 and 2) are rolled. Let's define the following events $$ A_{1} = \text{the result of dice 1 is even}\\ A_{2} = \text{the result of dice 2 is even} \\ A_{3} = \text{both dice give the same result} \\ $$ I need to compute $P(A_3 | A_1 \cap A_2)$ and $P(A_{1} \cup A_{2} \cup A_{3}^{c})$. My calculations are the following $$ P(A_3 | A_1 \cap A_2) = \frac{P(A_3 \cap A_1 \cap A_2) }{ P(A_1 \cap A_2)} = \frac{3/36}{3^2/36} = 1/3 $$
\begin{align*} P(A_{1} \cup A_{2} \cup A_{3}^{c}) & = 1 - P(A_1^c \cap A_2^c \cap A_3) \\ & = 1 - P(A_3| A_2^c \cap A_1^c)P(A_2^c \cap A_1^c) \\ &= 1 - \frac{1}{3} \cdot \frac{3^2}{36} \\ & = \frac{11}{12} \end{align*} but I don't know if it's correct. Could someone corroborate or help me?
They are correct, if you are in doubt you can just write explicitly the events and you can compare them to the results.