A fair die is rolled 4 times. What is the probability of having no 4 and no 5 among the rolls? Round your answer to three decimal places.

To solve this problem, let's determine the probability of not rolling a 4 or 5 in a single roll, and then extend this to four rolls of the die.

### Step 1: Probability of not rolling a 4 or 5 on a single roll
A fair die has 6 faces, and the numbers 4 and 5 are two of these faces. Therefore, the probability of not rolling a 4 or 5 on a single roll is:

\[
P(\text{not 4 or 5 in one roll}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{6} = \frac{2}{3}
\]

### Step 2: Probability of not rolling a 4 or 5 in 4 rolls
Since each roll is independent, the probability of not rolling a 4 or 5 in all 4 rolls is:

\[
P(\text{not 4 or 5 in 4 rolls}) = \left(\frac{2}{3}\right)^4
\]

### Step 3: Calculate the probability
\[
P(\text{not 4 or 5 in 4 rolls}) = \left(\frac{2}{3}\right)^4 = \frac{16}{81} \approx 0.1975
\]

### Step 4: Round to three decimal places
\[
P(\text{not 4 or 5 in 4 rolls}) \approx 0.198
\]

Thus, the probability of having no 4 and no 5 among the rolls is approximately **0.198**.

Would you like further details or explanations? Here are some related questions:

1. What is the probability of rolling at least one 4 or 5 in 4 rolls?
2. How would the probability change if the die is rolled 5 times instead?
3. What is the probability of rolling exactly two 4s or 5s in 4 rolls?
4. What is the probability of rolling a 6 exactly once in 4 rolls?
5. How does the probability change if the die is biased?
6. What is the expected number of times a 4 or 5 appears in 4 rolls?
7. Can you calculate the variance of the number of 4s or 5s in 4 rolls?
8. What is the probability of rolling four consecutive even numbers?

**Tip:** When dealing with independent events, multiplying their probabilities gives you the probability of all events occurring together.

Learn how to calculate the probability of not rolling a 4 or 5 in four rolls of a fair six-sided die. Step-by-step solution provided with detailed calculations and rounding to three decimal places. Suitable for students studying probability in Grades 9-12.

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