Question

Imagine that you roll two dice. Write down all the possible rolls that sum to \(2 .\) Write all the possible rolls that sum to \(12 .\) Write all the possible rolls that sum to 7 . Which configuration has the greatest entropy: \(2,12,\) or \(7 ?\)

Short answer

The possible roll that sums to 2 is (1,1). The possible roll that sums to 12 is (6,6). The possible rolls that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). The configuration with sum of 7 has the greatest entropy.

Step-by-step solution

Identify Possibilities for Sum of 2

To get a sum of 2, the only possibility is when both dice show 1 (since the lowest value on a die is 1). So the only possible roll is (1,1).

Identify Possibilities for Sum of 12

To get a sum of 12, both dice need to show the maximum value which is 6. So the only possible roll for a sum of 12 is (6,6).

Identify Possibilities for Sum of 7

A sum of 7 can occur with more combinations because it is in the middle of the range of possible sums (2-12). The possible rolls are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

Compare Entropy

Entropy refers to the degree of disorder or randomness in a system. The configuration with the greatest number of possible outcomes has the greatest entropy. Hence, sum of 7 has the greatest entropy with 6 different possible rolls, compared to the sums of 2 and 12, which only have 1 possible roll each.

Key concepts

Probabilistic Outcomes

When we speak about probabilistic outcomes, we are referring to the different results that can happen when an event takes place. For example, when rolling a single die, there are six possible outcomes because the die can land on any one of its six faces.

In the context of rolling two dice, as mentioned in the exercise, the total number of probabilistic outcomes is the product of the individual outcomes of each die. Since each die can land on one of six faces, the two dice together have a total of 6 multiplied by 6, which equals 36 different possible outcomes. This calculation allows us to understand the likelihood of different sums being rolled. For instance, the sum of 2 can only be achieved by both dice showing a 1, making it a less likely event than rolling a sum of 7, which can be achieved with six different dice combinations.

Entropy and Randomness

The concept of entropy is associated with the amount of disorder or randomness in a system. In the realm of probability, a system with high entropy is one in which there are many possible states or outcomes that can occur, making the outcome harder to predict. In contrast, a system with low entropy has fewer possible states and the outcome can be predicted with greater certainty.

In the dice rolling example, a sum of 7 presents the highest entropy among the given sums because it has six different combinations that can produce it (such as (1,6), (2,5), and (3,4)), while sums of 2 and 12 each only have one combination (1,1) and (6,6) respectively. High entropy in this case signifies a higher level of randomness or unpredictability in the result.

Dice Roll Probability

When calculating the probability of rolling a certain sum with two dice, it's important to understand that not all sums have the same probability of occurring. A sum of 7, as the exercise shows, has the highest probability as there are six different ways to roll it. Since there are 36 possible outcomes when rolling two dice, the probability of rolling a 7 is 6 out of 36, which can be simplified to 1 out of 6 or approximately 16.67%.

By contrast, the sums of 2 and 12 are less probable because there is only one combination to achieve each of these sums ((1,1) for 2, and (6,6) for 12). This means that the probability of rolling a 2 or a 12 is 1 out of 36, or about 2.78%. Remembering this makes it easier to understand why different sums are more or less likely when playing games of chance that involve dice.