Understanding the probability of events when rolling dice is a fundamental concept in probability theory. This problem asks for the probability of a specific outcome: rolling an even number on each of two identical dice rolled simultaneously.

Total Outcomes for Dice Rolls

When two identical cube-shaped dice are rolled simultaneously, each die has 6 possible outcomes (1, 2, 3, 4, 5, 6). To find the total number of possible outcomes for both dice, we multiply the number of outcomes for each individual die.

• Possible outcomes for the first dice = 6
• Possible outcomes for the second dice = 6

Therefore, the total number of possible outcomes (sample space) when rolling two dice is:

$$ \text{Total Outcomes} = 6 \times 6 = 36 $$

These 36 outcomes can be represented as ordered pairs, for example: (1,1), (1,2), ..., (6,6).

Even Numbers on a Single Dice

For a single dice, the faces are numbered 1, 2, 3, 4, 5, and 6. We need to identify the even numbers among these.

• The even numbers are 2, 4, and 6.

So, there are 3 favorable outcomes for rolling an even number on a single dice.

The probability of rolling an even number on a single dice is:

$$ P(\text{Even on one dice}) = \frac{\text{Number of even outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} $$

Probability of Even Numbers on Both Dice

Since the two dice rolls are independent events (the outcome of one dice does not affect the outcome of the other), the probability that an even number is rolled out on each dice is the product of the probabilities of rolling an even number on each individual dice.

Let $E_1$ be the event of rolling an even number on the first dice, and $E_2$ be the event of rolling an even number on the second dice.

We want to find $P(E_1 \text{ and } E_2)$.

Since $E_1$ and $E_2$ are independent events:

$$ P(E_1 \text{ and } E_2) = P(E_1) \times P(E_2) $$

We already calculated $P(E_1) = \frac{1}{2}$ and $P(E_2) = \frac{1}{2}$.

Therefore:

$$ P(\text{Even on each dice}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} $$

Alternative Method: Favorable Outcomes for Both Dice

Alternatively, we can list all the favorable outcomes where both dice show an even number:

• First dice shows an even number (2, 4, or 6).
• Second dice shows an even number (2, 4, or 6).

The combinations where both dice show an even number are:

There are 9 such favorable outcomes.

The probability is the ratio of favorable outcomes to the total possible outcomes:

$$ P(\text{Even on each dice}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{9}{36} $$

Simplifying the fraction:

$$ \frac{9}{36} = \frac{1}{4} $$

Conclusion on Dice Probability

Both methods yield the same result. The probability that an even number is rolled out on each dice when two identical cube shaped dice are rolled simultaneously is $\frac{1}{4}$.