Probability of Drawing 2 Yellow Marbles with and without Replacement

When working with probability, understanding how to calculate the likelihood of specific events is fundamental. This article delves into the probability of drawing 2 yellow marbles from a bag containing 3 yellow marbles and 2 red marbles, both with and without replacement.

Understanding the Setup

Consider a bag with 3 yellow marbles and 2 red marbles. We need to calculate the probability of drawing 2 yellow marbles if the draws are made with or without replacement. Let's explore both scenarios step by step.

With Replacement

In this scenario, after drawing a marble, it is returned to the bag, ensuring that the probability of drawing a yellow marble remains constant for each draw.

Step 1: Calculate the Probability of Drawing a Yellow Marble on One Draw

The total number of marbles is 5, with 3 yellow and 2 red. The probability of drawing a yellow marble on one draw can be calculated as follows:

P(text{yellow}) frac{3}{5}

Step 2: Calculate the Probability of Drawing 2 Yellow Marbles with Replacement

Since the draws are made with replacement, the probability of drawing a yellow marble remains the same for both draws. Therefore:

P(text{2 yellow}) P(text{yellow}) times P(text{yellow}) frac{3}{5} times frac{3}{5} frac{9}{25}

To confirm this result, a brute force approach can be used in the J programming language to simulate 10,000,000 draw pairs. The result indicates a probability of approximately 0.36, which is consistent with our calculation.

Without Replacement

Now, let's consider the scenario where the first marble is not put back in the bag, changing the probability of drawing subsequent yellow marbles.

Step 1: Calculate the Probability of Drawing a Yellow Marble on the First Draw

The probability of drawing a yellow marble on the first draw remains:

P(text{first yellow}) frac{3}{5}

Step 2: Update the Bag and Calculate the Probability of Drawing a Yellow Marble on the Second Draw

After drawing a yellow marble on the first draw, there are now 2 yellow marbles and 2 red marbles left. The probability of drawing a yellow marble on the second draw is:

P(text{second yellow}) frac{2}{4} frac{1}{2}

Step 3: Calculate the Combined Probability

The combined probability of both events (drawing two yellow marbles in a row) without replacement is:

P(text{2 yellow, no replacement}) P(text{first yellow}) times P(text{second yellow}) frac{3}{5} times frac{1}{2} frac{3}{10}

Comparing Results

When comparing the results of drawing with and without replacement, we see that:

With replacement: P(text{2 yellow}) frac{9}{25} Without replacement: P(text{2 yellow}) frac{3}{10}

Both probabilities can be converted to decimal form for a more straightforward comparison:

With replacement: 0.36 Without replacement: 0.30

Conclusion

This exploration has shown the significant impact that the method of sampling (with or without replacement) has on the probability of drawing 2 yellow marbles. Understanding these concepts is crucial in various fields, including statistics and data analysis, as it allows for accurate predictions and interpretations of outcomes.

Experimenting and applying these principles with real-world data can further enhance your understanding of probability. Don't forget to leverage tools and programming languages like the J programming language to simulate and verify your findings.