If five fair, six-sided dice are rolled simultaneously, to the nearest thousandth, what is the probability of rolling no threes?

To determine the probability of rolling no threes when five fair six-sided dice are rolled, we first need to calculate the probability of rolling a die and not getting a three.

Each die has six faces, and only one of those faces shows a three. Therefore, the probability of not rolling a three on a single die is the number of favorable outcomes divided by the total number of outcomes, which is:

[

P(\text{not a three on one die}) = \frac{5}{6}

]

Since the dice rolls are independent events, the probability of not rolling a three on all five dice simultaneously is the product of the probabilities for each die:

[

P(\text{not a three on all five dice}) = \left(\frac{5}{6}\right)^5

]

Calculating this gives:

[

\left(\frac{5}{6}\right)^5 = \frac{5^5}{6^5} = \frac{3125}{7776} \approx 0.402

]

When rounded to the nearest thousandth, this result is indeed approximately 0.402, which corresponds with the choice provided. Thus, the correct answer reflects the likelihood of rolling