Understanding the Context

Understanding the Problem

Imagine a box containing a total of marbles — red and non-red (let’s say blue, for clarity). The goal is to calculate the likelihood of drawing exactly two red marbles in a sample, possibly under specific constraints like substitution, without replacement, or fixed total counts.

The precise probability value expressed as $oxed{\dfrac{189}{455}}$ corresponds to a well-defined setup where:
- The total number of marbles involves combinations of red and non-red.
- Sampling method (e.g., without replacement) matters.
- The number of red marbles drawn is exactly two.

But why is the answer $\dfrac{189}{455}$ and not a simpler fraction? Let’s explore the logic behind this elite result.

Key Insights

A Step-By-Step Breakdown

1. Background on Probability Basics

The probability of drawing a red marble depends on the ratio:

$$
P(\ ext{red}) = rac{\ ext{number of red marbles}}{\ ext{total marbles}}
$$

But when drawing multiple marbles, especially without replacement, we rely on combinations:

Final Thoughts

$$
P(\ ext{exactly } k \ ext{ red}) = rac{inom{R}{k} inom{N-R}{n-k}}{inom{R + N-R}{n}}
$$

Where:
- $R$ = total red marbles
- $N-R$ = non-red marbles
- $n$ = number of marbles drawn
- $k$ = desired number of red marbles (here, $k=2$)

2. Key Assumptions Behind $\dfrac{189}{455}$

In this specific problem, suppose we have:

• Total red marbles: $R = 9$
  - Total non-red (e.g., blue) marbles: $N - R = 16$
  - Total marbles: $25$
  - Draw $n = 5$ marbles, and want exactly $k = 2$ red marbles.