Understanding the Dice Probabilities

We are given two identical long dice. Each die has four rectangular faces labeled with scores: 2, 3, 5, and 7.

The dice cannot land on the square faces. This means each of the four labeled faces has an equal probability of being the top face when a die is thrown.

The probability of landing on any specific face (2, 3, 5, or 7) for a single die is $P(\text{face}) = \frac{1}{4}$.

Calculating Total Possible Outcomes

When two such dice are thrown together, the total number of possible outcomes is the product of the number of outcomes for each die.

Total Outcomes = (Number of faces on Die 1) $\times$ (Number of faces on Die 2)

Total Outcomes = $4 \times 4 = 16$.

Each of these 16 outcomes is equally likely because the individual face probabilities are equal.

Identifying Favorable Outcomes (Sum > 11)

We need to find the pairs of scores (Die 1, Die 2) whose sum is greater than 11.

Let's list the possible pairs systematically:

• If Die 1 = 2, the required score on Die 2 must be > 9 (impossible with faces {2, 3, 5, 7}).
• If Die 1 = 3, the required score on Die 2 must be > 8 (impossible).
• If Die 1 = 5, the required score on Die 2 must be > 6. The only possibility is Die 2 = 7. Pair: (5, 7).
• If Die 1 = 7, the required score on Die 2 must be > 4. Possible scores are Die 2 = 5 or Die 2 = 7. Pairs: (7, 5), (7, 7).

The pairs of scores resulting in a sum greater than 11 are (5, 7), (7, 5), and (7, 7).

Number of Favorable Outcomes = 3.

Determining the Probability

The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

Probability (Sum > 11) = $\frac{\text{Number of Favorable Outcomes}}{\text{Total Possible Outcomes}}$

Probability (Sum > 11) = $\frac{3}{16}$.