A die is thrown three times. The chance that the highest number shown on the die is 4 is p, then the value of `[1//p]` is where [.] represents greatest integer function is _________.

Explanation

To solve the problem of finding the probability \( p \) that the highest number shown on a die thrown three times is 4, we can break down the solution step by step.

### Step 1: Understanding the Problem
When a die is thrown three times, the possible outcomes are the numbers 1, 2, 3, 4, 5, and 6. We need to find the probability that the highest number rolled is exactly 4. This means that all rolls must be 4 or less, and at least one roll must be a 4.

### Step 2: Total Outcomes
The total number of outcomes when a die is thrown three times is:
\[ 6^3 = 216 \]

### Step 3: Outcomes Where the Highest Number is 4
To find the outcomes where the highest number is exactly 4, we can consider two cases:
1. **At least one die shows a 4**.
2. **All dice show numbers from the set {1, 2, 3, 4}**.

### Step 4: Counting Favorable Outcomes
1. **Calculating the total outcomes where all numbers are ≤ 4**:
- The possible outcomes for each die are {1, 2, 3, 4}, which gives us:
\[ 4^3 = 64 \]

2. **Calculating the outcomes where no die shows a 4**:
- The possible outcomes for each die are {1, 2, 3}, which gives us:
\[ 3^3 = 27 \]

3. **Calculating the outcomes where at least one die shows a 4**:
- We subtract the outcomes where no die shows a 4 from the total outcomes where all numbers are ≤ 4:
\[ \text{Outcomes with at least one 4} = 64 - 27 = 37 \]

### Step 5: Calculating the Probability \( p \)
Now, we can find the probability \( p \) that the highest number shown is 4:
\[ p = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{37}{216} \]

### Step 6: Finding \( \left\lfloor \frac{1}{p} \right\rfloor \)
Next, we need to calculate \( \frac{1}{p} \):
\[ \frac{1}{p} = \frac{216}{37} \]
Calculating this gives:
\[ \frac{216}{37} \approx 5.8378 \]
Taking the greatest integer function:
\[ \left\lfloor \frac{216}{37} \right\rfloor = 5 \]

### Final Answer
Thus, the value of \( \left\lfloor \frac{1}{p} \right\rfloor \) is:
\[ \boxed{5} \]