Probability Analysis of Even Sums and Sums Less Than 7 on Two Dice Rolls

In dice probability, understanding the scenarios involving even outcomes and sums less than 7 can provide valuable insights. This article delves into the intricacies of these probabilities when two fair number cubes are rolled.

Understanding the Problem

To find the probability of obtaining an even sum or a sum less than 7, we start by determining the total number of possible outcomes and the specific outcomes that satisfy the conditions.

Total Outcomes

When rolling two fair dice, each die has 6 faces. Thus, the total number of possible outcomes is easily calculated as:

6 times; 6 36

Step 1: Calculate Outcomes for Even Sums

The possible even sums when rolling two dice include 2, 4, 6, 8, 10, and 12. We count the combinations for each even sum:

Sum 2: 11 (1 combination) Sum 4: 13, 22, 31 (3 combinations) Sum 6: 15, 24, 33, 42, 51 (5 combinations) Sum 8: 26, 35, 44, 53, 62 (5 combinations) Sum 10: 46, 55, 64 (3 combinations) Sum 12: 66 (1 combination)

Adding these combinations yields the total number of outcomes for even sums:

1 3 5 5 3 1 18

Step 2: Calculate Outcomes for Sums Less Than 7

The possible sums less than 7 are 2, 3, 4, 5, and 6. We count the combinations for each of these sums:

Sum 2: 11 (1 combination) Sum 3: 12, 21 (2 combinations) Sum 4: 13, 22, 31 (3 combinations) Sum 5: 14, 23, 32, 41 (4 combinations) Sum 6: 15, 24, 33, 42, 51 (5 combinations)

Adding these combinations yields the total number of outcomes for sums less than 7:

1 2 3 4 5 15

Step 3: Calculate Overlap Outcomes

However, there are sums that are both even and less than 7. We need to count these overlaps:

Sum 2: 11 (1 combination) Sum 4: 13, 22, 31 (3 combinations) Sum 6: 15, 24, 33, 42, 51 (5 combinations)

Adding these overlap combinations yields:

1 3 5 9

Step 4: Apply Inclusion-Exclusion Principle

To find the total number of outcomes that meet either condition, we use the inclusion-exclusion principle:

Total favorable outcomes Even outcomes Sums less than 7 outcomes - Overlapping outcomes

Total 18 15 - 9 24

Step 5: Calculate the Probability

Finally, we calculate the probability of getting an even sum or a sum less than 7:

P(even sum or sum 7) 24/36 2/3

Conclusion

The probability of generating an even sum or a sum less than 7 with a single roll of two fair number cubes is:

2/3