Understanding the Probability of Drawing Two Green Marbles with Replacement

Introduction to the Problem

When dealing with probabilities, particularly in the context of sampling with replacement, the outcome of one event does not affect the outcome of subsequent events. This independence is crucial in determining the overall probability of specific outcomes. In this article, we will explore the probability of drawing two green marbles from a bag containing 10 green, 7 white, and 3 red marbles, with the condition that the first marble drawn is returned to the bag before drawing the second one.

Steps to Calculate the Probability

To calculate the probability of drawing two green marbles with replacement from the mentioned bag, follow these steps: Identify the Total Number of Marbles: Number of green marbles: 10 Number of white marbles: 7 Number of red marbles: 3 Calculate the Probability of Drawing a Green Marble on the First Drawing: The total number of marbles is 20 (10 7 3). The probability of drawing a green marble on the first attempt is:

P(Green) Number of green marbles / Total number of marbles 10 / 20 1 / 2

Replace the First Marble and Draw Again: After replacing the first marble, the total number of marbles remains 20. There are still 10 green marbles in the bag. Calculate the Probability of Drawing a Green Marble on the Second Drawing: The probability of drawing a green marble on the second attempt, given that the first marble was replaced, is the same as the first:

P(Green on second) Number of green marbles / Total number of marbles 10 / 20 1 / 2

Calculate the Combined Probability for Drawing Two Green Marbles: Since the draws are independent, the combined probability is the product of the individual probabilities:

P(Two Green) P(Green on first) × P(Green on second) 1 / 2 × 1 / 2 1 / 4

The probability of drawing two green marbles with replacement is 1/4, or 0.25.

Key Concepts in Probability and Replacement

1. Independent Events: Events are independent if the outcome of one event does not influence the outcome of another. In this scenario, drawing a green marble and then replacing it does not affect the probability for the next draw. 2. Replacing the Marble: Replacing the marble ensures that the total number of marbles remains constant, and the composition of the bag does not change between draws. 3. Probability Calculation: The probability of multiple independent events is found by multiplying their individual probabilities. This is a fundamental concept in probability theory and statistical analysis.

Conclusion

Understanding the probability of drawing two green marbles with replacement from a bag involves recognizing the independence of each draw and the effect of replacement on the total number of marbles. By following these steps and key concepts, we can accurately calculate the probability. This knowledge is valuable in various real-world applications, including quality control, lottery systems, and research studies.

Frequently Asked Questions

How many marbles are in the bag? What is the probability of drawing a green marble on the first draw? Why does the probability remain the same for the second draw? What is the combined probability of drawing two green marbles? The bag contains 20 marbles: 10 green, 7 white, and 3 red. The probability of drawing a green marble in the first draw is 1/2, and this probability remains the same for the second draw due to replacement. The combined probability of drawing two green marbles is 1/4 or 25%.