Probability of Drawing a Red Marble in Two Successive Draws: A Comprehensive Guide

Introduction:

In probability theory, understanding the basics of drawing marbles from a bag can provide a foundational insight into more complex probabilistic scenarios. This article explains the probability calculation for drawing a red marble from a bag containing six distinctly different colored marbles (red, blue, green, yellow, purple, and white), with a specific focus on the scenario where the first marble is replaced before the second draw.

Scenario:

Consider a bag containing 6 marbles: one red, one blue, one green, one yellow, one purple, and one white. In this scenario, we want to calculate the probability of drawing a red marble in the first draw and then drawing another red marble in the second draw, with the first marble being replaced.

Understanding the Basic Probability

The probability of drawing a red marble in the first draw is calculated as follows:

Step 1: Determine the total number of marbles in the bag. There are 6 marbles in total.

Step 2: Identify the favorable outcomes. There is only one red marble, making the favorable outcome 1.

Step 3: Compute the probability. The probability is the ratio of the favorable outcomes to the total outcomes:

Probability of drawing a red marble in the first draw 1 / 6

Replacement and the Second Draw

After the first marble is drawn, the marble is replaced back into the bag. This means that the total number of marbles in the bag remains 6, and the composition of the bag is exactly the same as before the first draw. Consequently, the probability of drawing a red marble in the second draw is the same as in the first draw:

Probability of drawing a red marble in the second draw 1 / 6

Calculating the Composite Probability

To find the probability of drawing a red marble in the first draw and then drawing another red marble in the second draw, the probabilities of the two events need to be multiplied:

The required probability (Probability of drawing a red marble in the first draw) (Probability of drawing a red marble in the second draw)

The required probability 1 / 6 1 / 6 1 / 36

Conclusion

Thus, the probability of drawing a red marble in the first draw and then drawing another red marble in the second draw, with the first marble being replaced, is 1/36. This simple example demonstrates the fundamental principles of probability and the importance of understanding the impact of replacement on the overall probability calculation.

Keywords: probability calculation, marble drawing, replacement in probability