Probability of Drawing a Red Marble After a Blue One Without Replacement

Understanding probability is crucial in many areas, including statistics and game theory. This article explores a specific scenario involving drawing marbles from a bag without replacement and investigates the probability of drawing a red marble after a blue one. The article aligns with Google's SEO standards by including relevant keywords, structured content, and comprehensible explanations.

Problem Description

The problem presented is: A bag contains 2 red and 6 blue marbles. What is the probability of picking a red marble after picking a blue one without putting back the marbles? This scenario requires calculating the probability of two sequential events under specific conditions.

Calculating the Probability

Let's break down the problem step-by-step:

Step 1: Drawing a Blue Marble First

The total number of marbles in the bag initially is 2 red 6 blue 8 marbles. The probability of drawing a blue marble first is:

P(Blue)

Step 2: Drawing a Red Marble Second

After drawing a blue marble, there are 7 marbles left in the bag, including 2 red marbles. The probability of drawing a red marble next is:

P(Red after Blue)

To find the combined probability of both events (drawing a blue marble first and then a red marble second), we multiply the probabilities of the two independent events:

P(Total) P(Blue) * P(Red after Blue) 0.75 *

Further Examples and Calculations

For further clarity, let's consider a more generalized example involving a bag with 4 blue, 3 yellow, and 7 red marbles. Here's how to calculate the probability in this case:

General Bag Example

The probability of drawing a blue marble first is:

P(Blue)

After drawing a blue marble, there are 13 marbles left in the bag, including 7 red marbles. The probability of drawing a red marble next is:

P(Red after Blue)

The combined probability is found by multiplying the probabilities of the two events:

P(Total) P(Blue) * P(Red after Blue)

Conclusion

The concept of probability in such scenarios is foundational for understanding more complex statistical and probabilistic models. By breaking down the events and calculating the probabilities step-by-step, we can accurately determine the likelihood of a sequence of events occurring. This example and its calculations can be applied to various real-world situations, such as in game theory, decision-making processes, and risk analysis.

Keywords: probability calculation, marbles probability, blue red marble