Let's determine the theoretical probability of rolling a 1 or a 6 on a standard number cube (a six-sided die).
### Step-by-Step Solution:
1. Identify the total number of possible outcomes:
A standard number cube has 6 faces, each showing a different number from 1 to 6. So, the total number of possible outcomes when you roll the die is 6.
2. Identify the favorable outcomes:
We are interested in rolling a 1 or a 6. These two outcomes are the favorable outcomes for our question. There are 2 favorable outcomes (rolling a 1 or a 6).
3. Determine the probability formula:
The theoretical probability [tex]\( P \)[/tex] of an event is given by:
[tex]\[
P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\][/tex]
4. Apply the numbers:
Here, the number of favorable outcomes is 2 (rolling a 1 or a 6), and the total number of possible outcomes is 6 (all faces of the die).
Substituting these into the probability formula, we get:
[tex]\[
P(\text{1 or 6}) = \frac{2}{6}
\][/tex]
5. Simplify the fraction:
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:
[tex]\[
P(\text{1 or 6}) = \frac{2 \div 2}{6 \div 2} = \frac{1}{3}
\][/tex]
### Conclusion:
The theoretical probability of rolling a 1 or a 6 on a standard number cube is [tex]\( \frac{1}{3} \)[/tex].
