You roll a die three times. What is the probability that it shows a one everyroll?

1. Break Down the Problem

To find the probability of rolling a one three times in a row with a fair six-sided die, we need to understand the individual probability of rolling a one and then apply it to the three rolls.

2. Relevant Concepts

The probability of rolling a one on a six-sided die is given by: $P(1) = \frac{1 \text{ favorable outcome}}{6 \text{ total outcomes}} = \frac{1}{6}$ Since the rolls are independent, the probability of rolling a one three times in a row is: $P(\text{three ones}) = P(1) \times P(1) \times P(1) = \left(\frac{1}{6}\right)^3$

3. Analysis and Detail

Now we calculate: $P(\text{three ones}) = \left(\frac{1}{6}\right)^3 = \frac{1}{6 \times 6 \times 6} = \frac{1}{216}$

4. Verify and Summarize

We arrived at the conclusion by checking the independence of the events and ensuring that the calculations were consistent. Therefore, the probability of rolling a one on three consecutive rolls of a six-sided die is $\frac{1}{216}$.

Final Answer

The probability that the die shows a one on every roll when rolled three times is $\frac{1}{216}$.