Probability Even Numbers Dice Calculation

The question asks for the probability of rolling an even number on each of two identical dice rolled simultaneously.

Understanding Dice Outcomes

Each standard die has 6 faces, numbered 1 to 6. The total possible outcomes for a single die roll are {1, 2, 3, 4, 5, 6}.

• Total possible outcomes for one die: 6

Identifying Favorable Outcomes

We are interested in rolling an even number. The even numbers on a die are {2, 4, 6}.

• Number of even outcomes for one die: 3

Calculating Single Die Probability

The probability of rolling an even number on a single die is the ratio of favorable outcomes to total outcomes.

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

Probability for Two Dice

Since the two dice are rolled simultaneously, the events are independent. The probability of independent events occurring together is the product of their individual probabilities. We need an even number on the first die AND an even number on the second die.

• $P(\text{Even on Die 1} \text{ and } \text{Even on Die 2}) = P(\text{Even on Die 1}) \times P(\text{Even on Die 2})$
• $P(\text{Both Even}) = \frac{1}{2} \times \frac{1}{2}$
• $P(\text{Both Even}) = \frac{1}{4}$

Conclusion

The probability that an even number is rolled on each die is $\frac{1}{4}$.