Question:
An unbiased dice with faces marked \(1,2,3,4,5,6\) is rolled four times. Out of four 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
An unbiased dice with faces marked \(1,2,3,4,5,6\) is rolled four times. Out of four 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
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When every trial has restricted allowed outcomes, compute probability using:
\[
\left(\frac{\text{allowed outcomes}}{\text{total outcomes}}\right)^n
\]
where \(n\) is the number of trials.
Updated On: Jun 26, 2026
- \(\dfrac{16}{81}\)
- \(\dfrac{1}{81}\)
- \(\dfrac{80}{81}\)
- \(\dfrac{65}{81}\)
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The Correct Option is A
Solution and Explanation
Step 1: Understand the given condition.
The minimum face value is not less than \(2\).
So, no outcome can be \[ 1 \] Also, the maximum face value is not greater than \(5\).
So, no outcome can be \[ 6 \]
Step 2: Identify the allowed face values.
Hence, each roll can contain only \[ 2,3,4,5 \] Thus, for each roll there are \[ 4 \] allowed outcomes.
Step 3: Find the total number of unrestricted outcomes.
Since the die is rolled \(4\) times and each roll has \(6\) outcomes, \[ \text{Total outcomes}=6^4 \] \[ =1296 \]
Step 4: Find the favorable outcomes.
For each of the \(4\) rolls, there are \(4\) acceptable values: \[ 2,3,4,5 \] Therefore, \[ \text{Favorable outcomes}=4^4 \] \[ =256 \]
Step 5: Compute the probability.
\[ \text{Probability} = \frac{4^4}{6^4} \] \[ = \left(\frac{4}{6}\right)^4 \] \[ = \left(\frac{2}{3}\right)^4 \] \[ = \frac{16}{81} \]
Step 6: Match with the options.
The obtained probability is \[ \frac{16}{81} \]
Step 7: Final conclusion.
Therefore, \[ \boxed{\frac{16}{81}} \]
The minimum face value is not less than \(2\).
So, no outcome can be \[ 1 \] Also, the maximum face value is not greater than \(5\).
So, no outcome can be \[ 6 \]
Step 2: Identify the allowed face values.
Hence, each roll can contain only \[ 2,3,4,5 \] Thus, for each roll there are \[ 4 \] allowed outcomes.
Step 3: Find the total number of unrestricted outcomes.
Since the die is rolled \(4\) times and each roll has \(6\) outcomes, \[ \text{Total outcomes}=6^4 \] \[ =1296 \]
Step 4: Find the favorable outcomes.
For each of the \(4\) rolls, there are \(4\) acceptable values: \[ 2,3,4,5 \] Therefore, \[ \text{Favorable outcomes}=4^4 \] \[ =256 \]
Step 5: Compute the probability.
\[ \text{Probability} = \frac{4^4}{6^4} \] \[ = \left(\frac{4}{6}\right)^4 \] \[ = \left(\frac{2}{3}\right)^4 \] \[ = \frac{16}{81} \]
Step 6: Match with the options.
The obtained probability is \[ \frac{16}{81} \]
Step 7: Final conclusion.
Therefore, \[ \boxed{\frac{16}{81}} \]
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