A fair die is rolled 4 times. What is the probability of having number 2 and number 6 among the rolls?

We are tasked with finding the probability of rolling both a 2 and a 6 in four rolls of a fair die. Let’s break this down step-by-step:

### Total Outcomes:
Since the die is rolled 4 times and each roll has 6 possible outcomes (numbers 1 through 6), the total number of outcomes is:

\[
6^4 = 1296
\]

### Favorable Outcomes:
To have both a 2 and a 6 among the rolls, let's consider the possible ways this can happen. We must choose two positions (out of the 4 rolls) to place the numbers 2 and 6, and then fill the remaining two positions with any of the other numbers (except for 2 and 6).

1. **Choosing positions for 2 and 6:** We can choose 2 positions from the 4 rolls for the numbers 2 and 6. The number of ways to choose 2 positions from 4 is given by the combination:

\[
\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
\]

2. **Arranging 2 and 6 in the chosen positions:** There are 2 ways to arrange the numbers 2 and 6 in the selected positions (since we can put 2 in one of the positions and 6 in the other, or vice versa).

So, the total number of ways to place 2 and 6 is:

\[
6 \times 2 = 12
\]

3. **Filling the remaining 2 positions:** The remaining two positions can be filled with any number from 1, 3, 4, or 5 (since 2 and 6 are already used). Each of these two positions has 4 possible choices (numbers 1, 3, 4, or 5), so the total number of ways to fill these positions is:

\[
4 \times 4 = 16
\]

### Total favorable outcomes:
Thus, the total number of favorable outcomes is:

\[
12 \times 16 = 192
\]

### Probability:
Finally, the probability of having both a 2 and a 6 among the 4 rolls is:

\[
\frac{192}{1296} = \frac{1}{6}
\]

So, the probability of having both a 2 and a 6 among the rolls is:

\[
\boxed{\frac{1}{6}}
\]

Would you like more details on this calculation or further explanation?

Here are some related questions for you to explore:
1. What is the probability of rolling at least one 6 in four rolls?
2. What is the probability of rolling all different numbers in four rolls?
3. What is the probability of rolling exactly two 6's in four rolls?
4. How many outcomes result in rolling no 2’s or 6’s in four rolls?
5. What is the expected number of times a 2 is rolled in four rolls?

**Tip:** When calculating probabilities, start by identifying the total number of possible outcomes and then focus on finding favorable outcomes.

Learn how to find the probability of rolling both a 2 and a 6 in four rolls of a fair die. This problem explores probability concepts and combinatorics, covering key steps like calculating total outcomes, determining favorable outcomes, and using the probability formula. Suitable for students in Grades 9-12.

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