If a fair die is rolled 15 times, what is the probability of rolling a 5 at least once, rounded to the nearest thousandth?

To determine the probability of rolling a 5 at least once when a fair die is rolled 15 times, we first need to understand the complementary approach. Instead of calculating the probability directly of rolling a 5 at least once, it is easier to first find the probability of not rolling a 5 at all in 15 rolls, and then subtract that value from 1.

The probability of not rolling a 5 in a single throw of a fair die is ( \frac{5}{6} ), because there are five other outcomes (1, 2, 3, 4, and 6).

When the die is rolled 15 times, the probability of not rolling a 5 in any of those rolls is calculated as follows:

[

\left( \frac{5}{6} \right)^{15}

]

Now, we compute that value:

[

\left( \frac{5}{6} \right)^{15} \approx 0.207

]

Next, we find the probability of rolling a 5 at least once by calculating:

[

1 - \left( \frac{5}{6} \right)^{15} \approx 1 - 0