Understanding the Probability of Drawing Marbles Without Replacement

In the field of probability, understanding how to calculate the chances of drawing specific items from a set, such as marbles, is a fundamental concept. This article explores the probability of drawing a blue marble followed by a red marble from a bag containing a specific number of each. We will break down the problem step-by-step, illustrate the solution, and provide additional resources for those interested in delving deeper into probability theory.

Introduction to the Problem

Imagine you have a bag containing 4 red marbles and 5 blue marbles. You want to calculate the probability of two events occurring in sequence: first, drawing a blue marble, and then, without replacing it, drawing a red marble. This is a classic example of a problem involving combinatorial probability, which is the study of probability related to the number of possible combinations or outcomes of an event.

Calculating the Probability

To solve this problem, we need to follow a systematic approach. The probability of the two events happening consecutively is the product of the probabilities of each individual event.

Step 1: Calculate the Total Number of Marbles

The total number of marbles in the bag is the sum of red and blue marbles:

4 (red) 5 (blue) 9 marbles

Step 2: Probability of Drawing a Blue Marble First

The probability of drawing a blue marble on the first attempt is the number of blue marbles divided by the total number of marbles: [ P(B_1) frac{5}{9} ]

Step 3: Probability of Drawing a Red Marble Second

After drawing a blue marble, there are now 8 marbles left in the bag (4 red and 4 blue). The probability of drawing a red marble now is: [ P(R_2 | B_1) frac{4}{8} frac{1}{2} ]

Step 4: Combined Probability

To find the combined probability of both events happening (drawing a blue marble first and then a red marble), we multiply the probabilities of each individual event: [ P(B_1 text{ and } R_2) P(B_1) times P(R_2 | B_1) frac{5}{9} times frac{1}{2} frac{5}{18} ] Therefore, the probability of randomly selecting a blue marble first and then, without replacing it, selecting a red marble is (frac{5}{18}).

Additional Insights and Resources

For those interested in further exploring the realm of probability and combinatorial probability, the following resources are highly recommended: - Quora Profile: Delve into more complex probability problems and understand various scenarios involving multiple events and outcomes.- Google Inside Search Blog: Stay updated with the latest SEO tips and tricks, including how to optimize your content for better ranking in search results.- Google Developers Community: Join the community and engage with fellow developers for discussions and best practices on web development and coding.

By understanding the concept of probability and its applications, you can tackle similar problems and enhance your problem-solving skills in various fields. Happy exploring!