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When working with probability models involving dice, it's crucial to understand the concept of total possible outcomes. Each die has 6 faces, numbered from 1 to 6. When rolling two dice, each face of the first die can pair with any face of the second die.
Therefore, the total number of outcomes is found by multiplying the number of faces on each die:
6 faces (die 1) × 6 faces (die 2) = 36 total outcomes.
This means, there are 36 different results when you roll two dice, ranging from (1,1) to (6,6).

Understanding the total possible outcomes is the foundation for calculating any probability involving dice. Make sure to count all combinations correctly to avoid errors similar to Bob's mistake.

When rolling two dice, each pair of numbers will result in a specific sum. These sums range from 2 (1+1) to 12 (6+6). However, not all sums appear with the same frequency.
For example:

• The sum of 2 can only occur in one way: (1,1).
• Similarly, sum of 3 can occur in two ways: (1,2) and (2,1).

The most common sum is 7, which can be achieved in six different ways: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Hence, each sum has different probabilities of occurring. Recognizing these combinations helps in understanding why some sums are more frequent than others.

To calculate the probability of a specific event occurring, divide the number of ways that event can happen by the total number of possible outcomes.
In the case of rolling a sum of 2 with two dice:

• Number of ways to get a sum of 2: 1 (only (1,1))
• Total possible outcomes: 36

The probability of rolling a sum of 2 is therefore \(\frac{1}{36}\). Correctly calculating probabilities relies on an accurate count of successful outcomes and total outcomes.

Combinatorial probability involves counting possible combinations and understanding how they relate to probability. In the context of dice rolls, it helps determine the likelihood of specific sums.
For example, to find the probability of rolling a sum of 7:

• Identify all possible combinations that give a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
• Count the number of successful combinations: 6.
• Total possible outcomes: 36.

The probability of rolling a sum of 7 is \(\frac{6}{36} = \frac{1}{6}\).
Combinatorial probability requires careful counting of possible successful outcomes, ensuring none are missed, and understanding their relation to the total number of possible outcomes.