Thus, the probability is $\boxed{\dfrac25216}$ - Simpleprint

Understanding the Context

What Is Probability?

At its core, probability measures the ratio of favorable outcomes to total possible outcomes in a well-defined sample space. For example, flipping a fair coin has two outcomes (heads or tails), each equally likely, giving a probability of $ \dfrac{1}{2} $. However, many probability problems involve more complex events with multiple stages, requiring careful counting and fraction simplification.

Context: Why $\dfrac{25}{216}$?

Key Insights

The fractional result $ \dfrac{25}{216} $ commonly appears in problems involving independent events across multiple trials. A typical scenario involves rolling, sampling, or distributing events where multiple independent choices or repeated experiments alter the overall likelihood.

Step-by-Step Derivation: How We Get $\dfrac{25}{216}$

Step 1: Define the System
Suppose a situation involves three independent stages, each with a specific number of possible outcomes. For instance, imagine a game where three independent stages each involve choosing from 6 equally likely options. Though each stage has 6 choices, suppose certain outcomes combine with constrained probabilities.

Step 2: Calculate Total Outcomes
With three independent steps and 6 options per step, the total number of possible outcomes is:
$$
6 \ imes 6 \ imes 6 = 216
$$
This forms the denominator—representing all possible combinations.

Final Thoughts

Step 3: Identify Favorable Outcomes
Now, suppose the event of interest requires selecting exactly 2 specific outcomes from a subset of 25 favorable combinations across all possible trials. For example, only 25 out of the 216 total paths meet a precise condition (e.g., specific color-colors in a coloring problem or a bivariate probability scenario).

Step 4: Form the Probability Fraction
The probability is the ratio:
$$
\frac{\ ext{Number of favorable outcomes}}{\ ext{Total outcomes}} = \frac{25}{216}
$$

This fraction is already in simplest form since 25 and 216 share no common divisors other than 1.

Why This Mattered: Real-World Applications

Understanding why probabilities yield such exact fractions helps in fields ranging from risk analysis to game design:
- Risk Assessment: Calculating failure probabilities across multiple independent systems.
- Statistical Sampling: Estimating outcomes from survey data with multinomial distributions.
- Game Theory: Modeling complex turn-based games with sequential decisions.