An unbiased die is rolled 4 times. Out of 4 face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is

Options

A. `(16)/(81)`
B. `(1)/(81)`
C. `(80)/(81)`
D. `(65)/(81)`

Answer

To solve the problem of finding the probability that the minimum face value obtained from rolling an unbiased die 4 times is not less than 2 and the maximum face value is not greater than 5, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Range of Values**:
The minimum face value must be at least 2, and the maximum face value must be at most 5. Therefore, the possible face values we can obtain from the die rolls are 2, 3, 4, and 5.

2. **Determine the Total Outcomes**:
Since a die has 6 faces and is rolled 4 times, the total number of outcomes when rolling the die 4 times is:
\[ 6^4 = 1296 \]

3. **Identify the Favorable Outcomes**:
We need to count the outcomes where all rolled values are between 2 and 5 (inclusive). The valid outcomes are 2, 3, 4, and 5, which gives us 4 possible values.

4. **Calculate the Number of Favorable Outcomes**:
The number of outcomes where the face values are either 2, 3, 4, or 5 can be calculated as:
\[ 4^4 = 256 \]
This is because for each of the 4 rolls, there are 4 choices (2, 3, 4, or 5).

5. **Calculate the Required Probability**:
The probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{256}{1296} \]

6. **Simplify the Probability**:
To simplify \(\frac{256}{1296}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 256 and 1296 is 16:
\[ P = \frac{256 \div 16}{1296 \div 16} = \frac{16}{81} \]

### Final Answer:
Thus, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is:
\[ \frac{16}{81} \]