You are going to roll three dice. What is the probability that the highest of the three numbers will be exactly a 4? Please give your answer to three decimals without using a calculator.

To find the probability that the highest of three numbers rolled on three fair dice will be exactly 4, we need to analyze the possible outcomes. The highest number can be 4 only if each die rolls a number between 1 and 4, inclusive, and at least one die must roll a 4.

### Step-by-Step Solution

1. **Calculate the total number of possible outcomes:**
   Since each die can roll a number from 1 to 6, there are:
   \[
   6 \times 6 \times 6 = 216 \text{ possible outcomes}
   \]

2. **Calculate the number of favorable outcomes:**
   The highest number must be exactly 4. Thus, each die can only roll one of the numbers 1, 2, 3, or 4. There are:
   \[
   4 \times 4 \times 4 = 64 \text{ possible outcomes where each die shows 1, 2, 3, or 4}
   \]
   However, we need to exclude the cases where the highest number is less than 4 (i.e., the highest number is 3, 2, or 1).

3. **Calculate the cases where the highest number is less than 4:**
   - **Highest number is 3:**
     Each die can roll 1, 2, or 3. There are:
     \[
     3 \times 3 \times 3 = 27 \text{ outcomes}
     \]
   - **Highest number is 2:**
     Each die can roll 1 or 2. There are:
     \[
     2 \times 2 \times 2 = 8 \text{ outcomes}
     \]
   - **Highest number is 1:**
     Each die can only roll 1. There is:
     \[
     1 \times 1 \times 1 = 1 \text{ outcome}
     \]

4. **Total number of outcomes where the highest number is less than 4:**
   \[
   27 + 8 + 1 = 36
   \]

5. **Calculate the favorable outcomes where the highest number is exactly 4:**
   Subtract the number of outcomes where the highest number is less than 4 from the total outcomes where each die shows 1, 2, 3, or 4:
   \[
   64 - 36 = 28
   \]

6. **Calculate the probability:**
   The probability is the number of favorable outcomes divided by the total number of possible outcomes:
   \[
   P = \frac{28}{216} = \frac{7}{54}
   \]

To three decimal places, the probability is:
\[
\boxed{0.130}
\]

Would you like more details or have any questions?

Here are 5 more questions you can ask:
1. How do you calculate the probability of rolling an even sum with two dice?
2. What is the probability of getting exactly two sixes when rolling five dice?
3. How do you find the standard deviation of a set of numbers?
4. Can you explain the concept of Bayes' theorem with an example?
5. How do you solve a quadratic equation using the quadratic formula?

Tip: When solving probability problems, it's helpful to consider complementary events and use the principle of inclusion-exclusion for more complex cases.

Explore the probability of rolling three dice and having the highest number exactly 4. Detailed step-by-step solution provided without using a calculator. Suitable for advanced high school students.

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