An unbiased die with faces marked 1, 2, 3, 4, 5 and 6 is rolled four times. Out of four face values

Q: Wait, does 'not less than 2' mean 2 is included?

Yes, absolutely! 'Not less than 2' is mathematically equivalent to $x \ge 2$. This means 2 is a perfectly safe and favorable outcome. Only the number 1 is excluded.

Q: Why do we multiply the probabilities instead of adding them?

We multiply because the rolls are independent events. The outcome of the first roll has no influence on the second, third, or fourth. When we need multiple independent events to all occur, we use the multiplication rule: $P(A \cap B) = P(A) \times P(B)$.

Q: Can we use the complement method here?

You could, but it would be significantly more complex. You would have to calculate the probability of the union of events where the minimum is 1 or the maximum is 6, which involves the Principle of Inclusion-Exclusion. The direct method we used is much cleaner and less prone to error!

Question

An unbiased die with faces marked $1, 2, 3, 4, 5$ and $6$ 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

Options:

(A) $16/81$

(B) $1/81$

(C) $80/81$

(D) $65/81$

Sigma Percentile · Accepted Answer

### The Beauty of Probability: A Die-Rolling Journey

Probability is not just about numbers; it is about understanding the hidden structure of chance. When we look at a problem like this, it is easy to feel overwhelmed by the sheer number of combinations possible when rolling a die four times.

But if we pause and look at the constraints, the complexity melts away, revealing a beautiful, simple logic.

### Analyzing the Setup

Imagine you are holding a standard, unbiased die. It has six faces: $\{1, 2, 3, 4, 5, 6\}$.

The problem asks us to roll this die four times and satisfy two conditions simultaneously: the minimum face value must be at least 2, and the maximum face value must be at most 5.

If the minimum value is not less than 2, it means we cannot roll a 1. If the maximum value is not greater than 5, it means we cannot roll a 6.

Suddenly, the entire sample space of six numbers is filtered down. We are left with a 'safe zone' of numbers: $\{2, 3, 4, 5\}$.

### The Safe Zone

For our conditions to hold true across all four rolls, every single roll must land within this safe zone. If even one roll lands on a 1 or a 6, the entire experiment fails.

For any single roll, we have four favorable outcomes: $\{2, 3, 4, 5\}$. The total number of possible outcomes for a single roll remains 6.

Therefore, the probability of success for one roll, $P(A)$, is the ratio of favorable outcomes to total outcomes:

$$P(A) = \frac{4}{6} = \frac{2}{3}$$

This is the probability that a single roll lands safely inside our restricted set.

### The Power of Independence

Now, we must consider the four rolls. The magic of probability lies in the concept of independence.

Because the die has no memory, the outcome of the first roll does not influence the second, third, or fourth. When we need a sequence of independent events to all succeed, we multiply their individual probabilities.

We need the first roll to be successful, AND the second, AND the third, AND the fourth. Mathematically, this is:

$$P = P(A) 	imes P(A) 	imes P(A) 	imes P(A) = P(A)^4$$

Substituting our value, we get:

$$P = \left( \frac{2}{3} ight)^4$$

### Final Calculation

Raising a fraction to a power is straightforward: we raise both the numerator and the denominator to that power.

$$\left( \frac{2}{3} ight)^4 = \frac{2^4}{3^4}$$

Calculating these powers, we find $2^4 = 16$ and $3^4 = 81$.

Our final probability is:

$$\mathbf{P = \frac{16}{81}}$$

It is a clean, elegant result that emerges from the simple act of filtering out the forbidden numbers and recognizing the independence of each roll. Always look for the constraints first, and the math will follow.

Animated Solution for Mathematics - Probability: An unbiased die with faces marked $1, 2, 3, 4, 5$ and $6$ 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

Select Answer:

Visualized Solution

The Experiment: Rolling a Die

Constraint 1: Minimum Value $\geq 2$

Constraint 2: Maximum Value $\leq 5$

Defining the Favorable Set

Probability of Success in One Roll

Simplifying the Probability

Four Independent Rolls

Probability of Four Rolls

Final Result

The Sigma Insight: Addition and Multiplication Theorems

The Beauty of Probability

Analyzing the Setup

The Safe Zone

The Power of Independence

Final Calculation

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