First, let’s examine the total number of possible outcomes when rolling two fair standard dice. Each die has 6 faces, so the total number of possible outcomes is the product of the number of faces on each die:
[tex]\[ 6 \times 6 = 36 \][/tex]
Next, let’s identify the outcomes that result in a sum of 10. By examining all the pair combinations that sum to 10, we have:
- (4, 6)
- (5, 5)
- (6, 4)
So, there are 3 outcomes that result in a sum of 10:
[tex]\[ (4, 6), (5, 5), (6, 4) \][/tex]
Now, let’s identify the outcomes that result in doubles (both dice show the same number). These pairs are:
- (1, 1)
- (2, 2)
- (3, 3)
- (4, 4)
- (5, 5)
- (6, 6)
Thus, there are 6 outcomes that result in doubles:
[tex]\[ (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) \][/tex]
Next, we must consider the overlap between these two sets. The pair (5, 5) appears in both sets (outcomes summing to 10 and doubles).
To find the total number of successful outcomes (either sum of 10 or doubles), we combine these sets while avoiding double-counting. The combined unique outcomes are:
[tex]\[ (4, 6), (5, 5), (6, 4), (1, 1), (2, 2), (3, 3), (4, 4), (6, 6) \][/tex]
Counting these, we get a total of 8 successful outcomes.
Therefore, the probability of rolling a sum of 10 or rolling doubles is:
[tex]\[ \frac{8}{36} \][/tex]
This fraction represents the number of successful outcomes divided by the total number of possible outcomes. The answer should be expressed in this fraction form and not reduced:
[tex]\[ \boxed{\frac{8}{36}} \][/tex]
