To find the probability of rolling a number less than 5 (which means rolling a 1, 2, 3, or 4) on a number cube (which has the numbers 1 through 6), we first calculate the probability of rolling a number less than 5.
There are 4 favorable outcomes (1, 2, 3, 4) out of 6 possible outcomes. Therefore, the probability \(P\) of rolling a number less than 5 is:
\[ P(\text{less than 5}) = \frac{4}{6} = \frac{2}{3} \]
Now, we want the probability of this happening three times in a row. Since the rolls are independent, we can multiply the probabilities together:
\[ P(\text{less than 5 three times in a row}) = P(\text{less than 5}) \times P(\text{less than 5}) \times P(\text{less than 5}) = \left(\frac{2}{3}\right)^3 \]
Calculating this gives:
\[ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \]
Now, let's find the fraction from the given responses:
- \( \frac{8278}{27} \): Not a simple form, unrelated
- \( \frac{232}{3} \): Not applicable, unrelated.
- \( \frac{212}{1} \): Unrelated.
- \( \frac{12161}{216} \): Not a match.
However, from our calculations, the correct answer for the probability of rolling a number less than 5 three times in a row is:
\[ \frac{8}{27} \]
None of the provided options directly match this. This might indicate a mistake in the options provided. Please double-check the options given. The calculated probability based on standard dice rolling conventions is \( \frac{8}{27} \).