Dice Probability: Even Numbers and Sum

We need to find the probability of rolling two identical fair dice such that both dice show even numbers AND their sum is 5 or greater.

Sample Space Analysis

• When rolling two six-faced dice, the total number of possible outcomes is $6 \times 6 = 36$.

Favorable Outcomes Calculation

First, identify outcomes where both dice show even numbers. The even numbers are {2, 4, 6}. The possible pairs are:

• (2, 2), (2, 4), (2, 6)
• (4, 2), (4, 4), (4, 6)
• (6, 2), (6, 4), (6, 6)

There are $3 \times 3 = 9$ such outcomes.

Next, from these 9 outcomes, identify those where the sum is 5 or greater ($\text{Sum} \ge 5$):

• (2, 2): Sum = 4 (Exclude)
• (2, 4): Sum = 6 (Include)
• (2, 6): Sum = 8 (Include)
• (4, 2): Sum = 6 (Include)
• (4, 4): Sum = 8 (Include)
• (4, 6): Sum = 10 (Include)
• (6, 2): Sum = 8 (Include)
• (6, 4): Sum = 10 (Include)
• (6, 6): Sum = 12 (Include)

There are 8 outcomes that satisfy both conditions.

Probability Calculation

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

Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{8}{36}$

Simplifying the fraction:

Probability = $\frac{8}{36} = \frac{2}{9}$

Converting the fraction to a decimal:

Probability $\approx 0.2222...$

Rounding to two decimal places, the probability is 0.22.

This value (0.22) lies between 0.21 and 0.23.