To find the probability of rolling a factor of 21 with a 6-sided die, we need to follow these steps:
1. **Identify the factors of 21.**
The factors of 21 are numbers that can multiply together to give 21. The factors of 21 are 1, 3, 7, and 21.
2. **Determine the possible outcomes on a 6-sided die.**
A standard 6-sided die has sides numbered from 1 to 6. The possible outcomes of a roll are therefore 1, 2, 3, 4, 5, or 6.
3. **Find the factors of 21 among the possible outcomes.**
Comparing the possible outcomes of a die roll with the factors of 21, we see that the numbers 1 and 3 are common. The number 7 and 21 are not present on a 6-sided die, and thus cannot be outcomes of a roll.
4. **Count the number of favorable outcomes.**
There are 2 favorable outcomes: 1 and 3, because they are both factors of 21 and can be rolled on a 6-sided die.
5. **Count the total number of possible outcomes.**
There are 6 possible outcomes on a 6-sided die because there are 6 faces.
6. **Calculate the probability.**
The probability, P, of rolling a factor of 21 is the ratio of the number of favorable outcomes to the total number of possible outcomes.
\[ P(\text{factor of } 21) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} \]
Substituting in the numbers, we get:
\[ P(\text{factor of } 21) = \frac{2}{6} \]
7. **Simplify the fraction (if necessary).**
The fraction \(\frac{2}{6}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
\[ P(\text{factor of } 21) = \frac{2 \div 2}{6 \div 2} = \frac{1}{3} \]
Therefore, the probability of rolling a factor of 21 on a 6-sided die is \(\frac{1}{3}\).
