A fair die is rolled 6 times. What is the probability that a 4 is obtained on at least one of the rolls? Round your answer to three decimal places.

To find the probability that a 4 is obtained on at least one of the six rolls of a fair die, it's easier to first calculate the probability of the complementary event (not rolling a 4 in any of the rolls) and then subtract this from 1.

### Steps to Solve:

1. **Calculate the probability of not rolling a 4 in a single roll**:
   - Since a fair die has 6 faces, the probability of not rolling a 4 on any single roll is:
   \[
   P(\text{not 4}) = \frac{5}{6}
   \]

2. **Calculate the probability of not rolling a 4 in all six rolls**:
   - The events are independent, so the probability of not rolling a 4 in all 6 rolls is:
   \[
   P(\text{not 4 in all 6 rolls}) = \left(\frac{5}{6}\right)^6
   \]

3. **Calculate the probability of rolling at least one 4**:
   - The probability of getting at least one 4 is the complement of not getting a 4 at all:
   \[
   P(\text{at least one 4}) = 1 - P(\text{not 4 in all 6 rolls})
   \]

Now, let's compute the numerical values.

\[
P(\text{not 4 in all 6 rolls}) = \left(\frac{5}{6}\right)^6 \approx 0.3349
\]

\[
P(\text{at least one 4}) = 1 - 0.3349 = 0.6651
\]

### Final Answer:
The probability that a 4 is obtained on at least one of the rolls is approximately **0.665**.

Would you like to delve deeper into any part of this calculation or explore related questions?

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Here are 8 questions you might be interested in:

1. What is the probability of not getting a 4 in any roll over 6 rolls?
2. How does the probability change if the die is rolled 10 times instead of 6?
3. What is the probability of rolling a 4 exactly twice in 6 rolls?
4. How would the probability change if using an 8-sided die?
5. What if you want the probability of rolling a 4 on exactly 3 of the rolls?
6. How can we extend this to find the probability of rolling a 4 on a biased die?
7. What is the expected number of times a 4 will appear in 6 rolls?
8. How does this probability compare to rolling at least one 3 in 6 rolls?

**Tip:** Understanding complementary events can simplify many probability problems by focusing on what doesn’t happen rather than what does.

Calculate the probability of rolling at least one 4 on a fair die in six rolls. Understand the concept of complementary events and how to apply them to find the desired probability. Learn step-by-step with examples and practical calculations.

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