A single die is rolled 3 times. Find the probability of rolling one 1 times. State answer as a decimal rounded to six decimal places.

To find the probability of rolling one '1' in three rolls of a die, we can use the binomial probability formula. The probability of rolling a '1' on a six-sided die is \( \frac{1}{6} \), and the probability of not rolling a '1' is \( \frac{5}{6} \). The possible scenarios where exactly one '1' is rolled in three rolls can be represented as: - Rolling a '1' on the first roll and not on the others: \( \frac{1}{6} \cdot \left(\frac{5}{6}\right)^2 \) - Rolling a '1' on the second roll: \( \left(\frac{5}{6}\right) \cdot \frac{1}{6} \cdot \left(\frac{5}{6}\right) \) - Rolling a '1' on the third roll: \( \left(\frac{5}{6}\right)^2 \cdot \frac{1}{6} \) The total number of combinations in which one can achieve this is given by \( \binom{3}{1} = 3 \). Therefore, we calculate: \[ P(X = 1) = \binom{3}{1} \left(\frac{1}{6}\right)^1 \left(\frac{5}{6}\right)^2 = 3 \cdot \frac{1}{6} \cdot \left(\frac{25}{36}\right) = 3 \cdot \frac{25}{216} = \frac{75}{216} \approx 0.347222 \] Rounded to six decimal places, the probability of rolling a '1' exactly one time when rolling a die three times is approximately **0.347222**.