To determine the probability of rolling a 4 or a 6 on a single roll of a fair six-sided die, we can follow these steps:
1. Understand the Basics of Probability:
- A fair die has 6 faces, each numbered from 1 to 6.
- Each face has an equal probability of landing face up.
2. Determine the Probability of Rolling a Specific Number:
- Since there are 6 faces, the probability of rolling any specific number (such as a 4 or a 6) is [tex]\( \frac{1}{6} \)[/tex].
3. Identify Mutually Exclusive Events:
- Rolling a 4 and rolling a 6 are mutually exclusive events, meaning these events cannot occur simultaneously (you cannot roll both a 4 and a 6 at the same time on a single die).
4. Calculate the Probability of Either Event Occurring:
- Since rolling a 4 and rolling a 6 are mutually exclusive events, we can add their individual probabilities together to find the probability of either event occurring.
- Probability of rolling a 4: [tex]\( \frac{1}{6} \)[/tex]
- Probability of rolling a 6: [tex]\( \frac{1}{6} \)[/tex]
- Combining these probabilities for mutually exclusive events gives:
[tex]\[
P(\text{4 or 6}) = P(\text{4}) + P(\text{6}) = \frac{1}{6} + \frac{1}{6}
\][/tex]
5. Perform the Addition:
- Add the probabilities:
[tex]\[
\frac{1}{6} + \frac{1}{6} = \frac{2}{6}
\][/tex]
6. Simplify the Fraction:
- Simplify [tex]\( \frac{2}{6} \)[/tex]:
[tex]\[
\frac{2}{6} = \frac{1}{3}
\][/tex]
Therefore, the probability of rolling a 4 or a 6 on a single roll of a fair die is [tex]\( \frac{1}{3} \)[/tex].
So, the correct choice is:
c) [tex]\( \frac{1}{3} \)[/tex]
