I'm fairly certain that the probability of both dice returning an even number is $1/4$.
I got this by saying that since these are independent events, with each die returning an even number being $1/2$, then the probability of both being even is $1/2 \times 1/2 = 1/4$.
Further, there are 36 outcomes, and all possible even number combinations are $(2, 2), (2, 4), (2, 6), (4, 4), (4, 6), (6, 6), (6, 4), (6, 2), (4, 2)$. There are nine of them and $9/36 = 1/4$
What I can't seem to get over, is that there are an equal number of odd and even numbers, so, why is the answer not $1/2$?
I know that it's not one half, but I can't explain why.
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$\begingroup$What I can't seem to get over, is that there are an equal number of odd and even numbers, so, why is the answer not 1/2 ?
Because they are not complementary events. There is another possibility.
The probability that both dice show even numbers is: $1/4$
The probability that both dice show odd numbers is: $1/4$
The probability that one die shows even and the other shows odd is: $1/2$
$\endgroup$ $\begingroup$Instead of thinking of pairs of outcomes, incorporate the two dice into one, more complicated probability space. It is true that there are equally many odd and even score for a single die. But, with two dice scores being viewed as one outcome to a new, more complicated experiment, we have an exhaustive and mutually exclusive list of four events: (die 1: odd, die 2: odd), (even, odd), (odd, even), and (even, even). These events are equiprobable, so each has probability $1/4$. Notice that the probability of having an even score on at least one die is not less (actually, greater) than $1/2$.
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