The Probability of Drawing Exactly Two Red Marbles Explained: $oxed{\dfrac{189}{455}}$

When analyzing probability in combinatorics, few questions spark as much curiosity as the chance of drawing exactly two red marbles from a mixed set. Whether in games, science, or statistical modeling, understanding these odds helps in making informed decisions. This article breaks down the scenario where the probability of drawing exactly two red marbles is exactly $\dfrac{189}{455}$ — and how this number is derived using fundamental probability principles.

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.

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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:

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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.

Then the probability becomes:

$$
P(\ ext{exactly 2 red}) = rac{ inom{9}{2} \ imes inom{16}{3} }{ inom{25}{5} }
$$

Let’s compute this step-by-step.

Calculate the numerator: