To determine the probability of rolling an odd number on a standard six-sided die (number cube), we first note that there are three odd numbers on a die: 1, 3, and 5. Thus, the probability \( P(\text{odd}) \) of rolling an odd number is:
\[ P(\text{odd}) = \frac{\text{Number of odd outcomes}}{\text{Total outcomes}} = \frac{3}{6} = \frac{1}{2} \]
You want to find the probability of rolling an odd number three times in a row. Since each roll is independent, we can multiply the probabilities of rolling an odd number (which is \( \frac{1}{2} \)) for each of the three rolls:
\[ P(\text{odd three times}) = P(\text{odd}) \times P(\text{odd}) \times P(\text{odd}) = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \]
Now, looking at the given choices:
- \( \frac{1}{216} \)
- \( \frac{1}{2} \)
- \( \frac{3}{2} \)
- \( \frac{1}{8} \)
The correct answer is:
\[ \frac{1}{8} \]
Thus, the answer is \( \frac{1}{8} \).