Understanding the Probability Question

This question asks for the probability of a specific event occurring when two standard six-sided dice are thrown. We are interested in the outcomes where the absolute difference between the numbers shown on the two dice is either 2 or 3.

To solve any probability problem, we typically follow these steps:

• Determine the total number of possible outcomes.
• Determine the number of favorable outcomes (those that satisfy the condition).
• Calculate the probability using the formula: Probability = (Favorable Outcomes) / (Total Outcomes).

Total Possible Outcomes When Throwing Two Dice

When a single standard die is thrown, there are 6 possible outcomes (1, 2, 3, 4, 5, 6). When two dice are thrown, the outcome is an ordered pair of numbers, one from each die. The total number of possible outcomes is the product of the number of outcomes for each die.

Total outcomes = Number of outcomes on Die 1 × Number of outcomes on Die 2

Total outcomes = \(6 \times 6 = 36\)

We can list all 36 possible outcomes as pairs (Die 1 result, Die 2 result):

• (1,1), (1,2), (1,3), (1,4), (1,5), (1,6)
• (2,1), (2,2), (2,3), (2,4), (2,5), (2,6)
• (3,1), (3,2), (3,3), (3,4), (3,5), (3,6)
• (4,1), (4,2), (4,3), (4,4), (4,5), (4,6)
• (5,1), (5,2), (5,3), (5,4), (5,5), (5,6)
• (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)

Identifying Favorable Outcomes (Difference is 2 or 3)

We need to find the pairs from the list above where the absolute difference between the two numbers is either 2 or 3.

Outcomes where the difference is 2:

We look for pairs \((a, b)\) where \(|a - b| = 2\).

• If the first die is 1, the second must be 3: (1,3).
• If the first die is 2, the second must be 4: (2,4).
• If the first die is 3, the second must be 1 or 5: (3,1), (3,5).
• If the first die is 4, the second must be 2 or 6: (4,2), (4,6).
• If the first die is 5, the second must be 3: (5,3).
• If the first die is 6, the second must be 4: (6,4).

The outcomes with a difference of 2 are: (1,3), (2,4), (3,1), (3,5), (4,2), (4,6), (5,3), (6,4). There are 8 such outcomes.

Outcomes where the difference is 3:

We look for pairs \((a, b)\) where \(|a - b| = 3\).

• If the first die is 1, the second must be 4: (1,4).
• If the first die is 2, the second must be 5: (2,5).
• If the first die is 3, the second must be 6: (3,6).
• If the first die is 4, the second must be 1: (4,1).
• If the first die is 5, the second must be 2: (5,2).
• If the first die is 6, the second must be 3: (6,3).

The outcomes with a difference of 3 are: (1,4), (2,5), (3,6), (4,1), (5,2), (6,3). There are 6 such outcomes.

The question asks for the probability that the difference is 2 or 3. Since an outcome cannot have a difference of both 2 and 3 simultaneously, these two sets of outcomes are mutually exclusive. Therefore, we can simply add the number of outcomes for each case to find the total number of favorable outcomes.

Total favorable outcomes = (Outcomes with difference 2) + (Outcomes with difference 3)

Total favorable outcomes = \(8 + 6 = 14\)

Calculating the Probability

Now we can calculate the probability using the formula:

\(\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)

\(\text{Probability} = \frac{14}{36}\)

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

\(\text{Probability} = \frac{14 \div 2}{36 \div 2} = \frac{7}{18}\)

Conclusion

The probability that the difference of the numbers on the two dice is 2 or 3 is \(\frac{7}{18}\).

Revision Table: Key Concepts for Dice Probability

Additional Information on Probability with Dice

Understanding the sample space is crucial for dice problems. Each outcome in the sample space of rolling two fair dice is equally likely, with a probability of \(\frac{1}{36}\).

When dealing with events involving sums or differences, listing the outcomes systematically helps avoid missing any possibilities. For sums, outcomes tend to cluster around the middle (sum of 7 is most likely). For differences, the smallest differences (0 and 1) tend to be more common than larger differences.

For example, let's look at the number of outcomes for each possible absolute difference:

• Difference is 0: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) - 6 outcomes
• Difference is 1: (1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5) - 10 outcomes
• Difference is 2: (1,3), (3,1), (2,4), (4,2), (3,5), (5,3), (4,6), (6,4) - 8 outcomes (verified)
• Difference is 3: (1,4), (4,1), (2,5), (5,2), (3,6), (6,3) - 6 outcomes (verified)
• Difference is 4: (1,5), (5,1), (2,6), (6,2) - 4 outcomes
• Difference is 5: (1,6), (6,1) - 2 outcomes

Summing the counts: \(6 + 10 + 8 + 6 + 4 + 2 = 36\), which matches the total number of outcomes, confirming our lists are complete for differences.