Calculating the Probability of Drawing All Black Balls from a Basket

In this article, we will explore the probability of drawing three black balls from a basket containing blue, black, and red balls. This example uses the concept of combinations to solve the problem and provides insights into the calculation process, which can be applied in various real-world scenarios. We will also discuss how to optimize content for Google search engines.

Introduction to the Problem

The basket contains 3 blue, 5 black, and 3 red balls. We are interested in calculating the probability of drawing 3 black balls at random from this basket. This type of problem is commonly encountered in probability theory and statistics, and understanding the underlying concepts can be very useful for students, researchers, and professionals in various fields.

Understanding the Basics of Probabilities

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

The probability of an event E is given by:

P(E) frac{text{Number of favorable outcomes}}{text{Total number of possible outcomes}}

Case Study: Drawing 3 Black Balls from the Basket

Let's break down the problem step by step:

Step 1: Determine the Total Number of Balls

The basket contains:

3 blue balls 5 black balls 3 red balls

Therefore, the total number of balls is:

3 5 3 11

Step 2: Calculate the Number of Ways to Choose 3 Black Balls

The number of ways to choose 3 black balls from 5 is given by the combination formula:

binom{5}{3} frac{5!}{3! (5-3)!} frac{5 times 4}{2 times 1} 10

Step 3: Calculate the Total Number of Ways to Choose 3 Balls from 11

The total number of ways to choose 3 balls from 11 is:

binom{11}{3} frac{11!}{3! (11-3)!} frac{11 times 10 times 9}{3 times 2 times 1} 165

Step 4: Calculate the Probability

Now we can calculate the probability:

P(text{all black}) frac{binom{5}{3}}{binom{11}{3}} frac{10}{165} frac{2}{33}

Final Answer: The probability that all three balls drawn are black is frac{2}{33}.

Conclusion

This example demonstrates the application of combinations in probability calculations. Understanding these concepts can help in solving more complex problems in statistics and probability theory. By breaking down the problem into manageable steps, we can effectively calculate probabilities and gain insights into event likelihood.

SEO Tips for Google Search

To optimize this content for Google search:

Use Title Tags and Meta Descriptions: Include Title: "Calculating the Probability of Drawing All Black Balls from a Basket" and a concise Meta Description "Learn how to calculate the probability of drawing three black balls from a basket with 3 blue, 5 black, and 3 red balls." to improve click-through rates. Keywords and Headers: Utilize headers (H1, H2, H3) to structure the content and ensure keywords are strategically placed throughout the text. Internal and External Links: Use relevant internal and external links to provide additional context and trust signals. Yoast SEO Plugin: If using WordPress, consider the Yoast SEO plugin to optimize title tags, meta descriptions, and improve readability.

By following these SEO tips and using the keywords and headers provided, your content will be well-structured and optimized for search engines like Google.