The question you are working on asks you to find the theoretical probability of rolling a number less than 10 on a 12-sided die. Let's break it down:

Step 1: Total outcomes

Since the die is 12-sided, there are 12 possible outcomes (numbers 1 through 12).

Step 2: Successful outcomes

The event you're interested in is rolling a number less than 10. So, the numbers that are less than 10 are: 1, 2, 3, 4, 5, 6, 7, 8, 9. That's 9 possible outcomes.

Step 3: Probability formula

The probability of an event is the ratio of successful outcomes to total outcomes:

$P(\text{less than 10}) = \frac{\text{successful outcomes}}{\text{total outcomes}} = \frac{9}{12}$

Step 4: Simplify the fraction

$P(\text{less than 10}) = \frac{9}{12} = \frac{3}{4}$

Thus, the theoretical probability is $\frac{3}{4}$.

Would you like more details or have any questions on this?

Here are five related questions:

1. What is the probability of rolling an even number on the 12-sided die?
2. How would you calculate the probability of rolling a number greater than 10?
3. How does the probability of rolling a specific number on a die differ from the probability of a range of numbers?
4. Can the probability be greater than 1? Why or why not?
5. How do you calculate probabilities when there are multiple events involved?

Tip: Always check if a fraction can be simplified to make calculations easier!