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Given that:
- A fair die is rolled three times.
- A 1 is considered "success".
- All other outcomes are "failures".
You know that a die has 6 faces numbered from 1 to 6. In this case, these outcomes are considered "failures":
[tex]\begin{gathered} 2 \\ 3 \\ 4 \\ 5 \\ 6 \end{gathered}[/tex]That is a total of 5 numbers considered "failures".
Since the events are independent, you can determine that the probability of 0 successes is:
[tex]P=\frac{5}{6}\cdot\frac{5}{6}\cdot\frac{5}{6}[/tex]Solving the Multiplications, you get:
[tex]P=\frac{5\cdot5\cdot5}{6\cdot6\cdot6}[/tex][tex]P=\frac{125}{216}[/tex]Hence, the answer is:
[tex]P=\frac{125}{216}[/tex]