To solve the problem of finding the probability that the sum of two six-sided dice (one blue and one green) is less than 10, we can proceed as follows:
1. Determine all possible outcomes:
Each die can land on any of 6 faces. Since there are two dice, the total number of possible outcomes is calculated as:
[tex]\[
6 \times 6 = 36
\][/tex]
Therefore, there are 36 possible outcomes when rolling both dice.
2. Calculate the number of favorable outcomes:
We need to count the number of outcomes where the sum of the numbers on the two dice is less than 10. Let's go through all possible sums:
- Sum = 2: (1,1) → 1 outcome
- Sum = 3: (1,2), (2,1) → 2 outcomes
- Sum = 4: (1,3), (2,2), (3,1) → 3 outcomes
- Sum = 5: (1,4), (2,3), (3,2), (4,1) → 4 outcomes
- Sum = 6: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes
- Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes
- Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes
- Sum = 9: (3,6), (4,5), (5,4), (6,3) → 4 outcomes
Adding these up:
[tex]\[
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 = 30
\][/tex]
Therefore, there are 30 favorable outcomes where the sum of the dice is less than 10.
3. Calculate the probability:
The probability [tex]\( P \)[/tex] is the ratio of the number of favorable outcomes to the total number of possible outcomes. Thus,
[tex]\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{30}{36}
\][/tex]
Simplifying this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 6, we get:
[tex]\[
P = \frac{30 \div 6}{36 \div 6} = \frac{5}{6}
\][/tex]
Therefore, the probability that the sum of the numbers on the two dice will be less than 10 is:
[tex]\[
\frac{5}{6}
\][/tex]
In conclusion, the probability that Sky will roll a sum less than 10 with two six-sided dice is [tex]\(\boxed{\frac{5}{6}}\)[/tex].
