Four standard six-sided dice are rolled. Find the probability that, for each pair of dice, the product of the two numbers rolled on the dice is a multiple of 4.
This seems like a problem that could be solved with linearity of expectation (even though there is a simple solution with small casework). I'm not very comfortable with expected value or probability in general, but I'd like practice.
I think the solution would proceed by finding the expected value of the number of products that are multiples of $4$ from two random, independent dice rolls. But this would just be the probability that there would be a multiple of $4$ from two dice rolls.
The first case is that a $4$ is rolled. This happens with probability $11/36$. The second case to consider is given that no $4$ is rolled, we find the probability that the two dice rolls result in an even number from both rolls. This happens with probability $(2/5)^2=4/25$. But when we add the two together and multiply by $\binom{4}{2}$, the answer is absurd.
What went wrong is either that there is no linearity here, or my probability calculations are wrong (or both). How can I fix this attempt?
To get 4 you either multiply two even numbers or an odd number by 4. If one dice is odd, then the rest must be 4. So the only two options are either all even or one odd and the rest are 4. The total probability is $\left(\tfrac{1}{2}\right)^2 + 4\cdot\tfrac{1}{2}\cdot\left(\tfrac{1}{6}\right)^3$.
What you are trying to count is the expected number of pairs of dice whose product is 4. But you are not interested in that, you are interested in the event that all pairs have this quality.