Probability of Rolling a Sum of at Least 10 with Two Fair Six-Sided Dice

In probability theory, determining the likelihood of specific outcomes in a random experiment is a fundamental concept. One intriguing question in this domain concerns the probability of rolling a sum of at least 10 with two fair six-sided dice. Let's explore this problem in detail, breaking down the steps and calculations involved.

Understanding the Total Outcomes

When rolling two fair six-sided dice, the total number of possible outcomes is:

$$ 6 times 6 36 $$

This result comes from the fact that each die has 6 faces, and each roll is independent of the other.

Outcomes Yielding Sums of 10, 11, and 12

To find the probability that the sum is at least 10, we need to list and count the outcomes that yield sums of 10, 11, and 12. Let's examine each case:

Sum of 10

The pairs that result in a sum of 10 are:

4, 6 5, 5 6, 4

There are 3 outcomes that yield a sum of 10.

Sum of 11

The pairs that result in a sum of 11 are:

5, 6 6, 5

There are 2 outcomes that yield a sum of 11.

Sum of 12

The only pair that results in a sum of 12 is:

6, 6

There is 1 outcome that yields a sum of 12.

Total Outcomes for Sums of At Least 10

Now, adding the favorable outcomes, we get:

$$ 3 text{ for 10} 2 text{ for 11} 1 text{ for 12} 6 text{ outcomes} $$

Probability Calculation

The probability P that the sum is at least 10 is given by the ratio of favorable outcomes to total outcomes:

$$ P(text{sum} geq 10) frac{text{Number of favorable outcomes}}{text{Total outcomes}} frac{6}{36} frac{1}{6} $$

Thus, the probability that the sum of two rolls of a fair six-sided die is at least 10 is (frac{1}{6}).

Conclusion

Through this detailed analysis, we have determined that the probability of rolling a sum of at least 10 with two fair six-sided dice is (frac{1}{6}).

It's important to note that this problem can be approached in several ways. For instance, as demonstrated in the alternative solutions provided, different combinations and counts can be used, but the final probability remains consistent.

Understanding these calculations is crucial for anyone interested in probability theory, particularly for applications in games, statistics, and more complex probabilistic models.