Probability of Dice Sums: Greater Than 5 or Less Than 4

The probability of obtaining a sum greater than 5 or less than 4 when rolling two six-sided dice can be calculated through a series of steps involving understanding the possible outcomes and combining the probabilities of each condition.

Identifying Possible Outcomes

When rolling two six-sided dice, there are a total of 36 possible outcomes, since each die has 6 faces, and the total number of combinations is 6 x 6 36.

Condition Analysis: Sum Greater Than 5 or Less Than 4

The first step is to identify the outcomes where the sum of the dice is either greater than 5 or less than 4.

Condition 1: Sum Greater Than 5

The valid sums greater than 5 are 6, 7, 8, 9, 10, 11, and 12. We need to count the number of outcomes for each of these sums:

Sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1) - 5 outcomes Sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - 6 outcomes Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2) - 5 outcomes Sum of 9: (3,6), (4,5), (5,4), (6,3) - 4 outcomes Sum of 10: (4,6), (5,5), (6,4) - 3 outcomes Sum of 11: (5,6), (6,5) - 2 outcomes Sum of 12: (6,6) - 1 outcome

The total number of outcomes where the sum is greater than 5 is 5 6 5 4 3 2 1 26.

Condition 2: Sum Less Than 4

The valid sums less than 4 are 2 and 3. We need to count the number of outcomes for each of these sums:

Sum of 2: (1,1) - 1 outcome Sum of 3: (1,2), (2,1) - 2 outcomes

The total number of outcomes where the sum is less than 4 is 1 2 3.

Combining the Conditions

Since the conditions for sums greater than 5 and sums less than 4 are mutually exclusive, we can simply add the probabilities of these two conditions:

Probability (sum greater than 5 or less than 4) 26 3 29 outcomes out of the total 36 possible outcomes.

Calculating the Probability

The probability is calculated as:

Probability Number of favorable outcomes / Total number of outcomes 29 / 36.

Thus, the probability of obtaining a sum greater than 5 or less than 4 when rolling two dice is 29 / 36.

Generalization to 'n' Dice

For a general case involving 'n' dice:

When rolling two dice (n2), the total number of possibilities is 6^2 36.

Using the formula, we can express the sum of probabilities for sums greater than 5 or less than 4:

(frac{3}{36} frac{5}{36} frac{29}{36})

This can be derived from the following calculations:

For sums less than 4: 2 outcomes (2, 3) - P(2 or 3) 3/36

For sums greater than 5: 26 outcomes (see detailed breakdown above) - P(>5) 26/36

The combined probability is 3/36 26/36 29/36.

Alternatively, since the case of sum giving less than 4 or more than 5 is the opposite of 4 or 5, we can also find the probability as 1 - (probability of 4 or 5) 1 - (7/36) 29/36.

Conclusion

By understanding the possible outcomes and combining the probabilities of the required conditions, we can accurately calculate the probability of obtaining a sum greater than 5 or less than 4 when rolling two six-sided dice, which is 29/36.