Calculating the Probability of Drawing Specific Balls from a Given Box
Probability is a fundamental concept in mathematics that helps us understand the likelihood of events occurring. One common scenario in probability involves drawing balls from a box, which is a useful subject for SEO optimization. In this article, we will break down the steps to calculate the probability of drawing a specific number of balls from a box containing 6 white balls and 9 black balls, taking 5 balls out in total. Advanced SEO techniques can help in improving the visibility of such informative content on search engines like Google. Let's dive into the detailed steps to find the solution.Introduction and Problem Statement
We are given a box that contains a total of 15 balls (6 white and 9 black). We need to determine the probability of drawing an even number of white balls or 3 black balls or 4 black balls when we randomly draw 5 balls. We will use combinatorial mathematics to solve this problem.
Step 1: Total Ways to Choose 5 Balls
The first step in solving this problem is to calculate the total number of ways to choose 5 balls from the 15 balls in the box. This can be done using the combination formula, which is given by:
Combination formula: ( {n choose k} frac{n!}{k!(n-k)!} )
For our case:
( {15 choose 5} frac{15!}{5!(15-5)!} frac{15 times 14 times 13 times 12 times 11}{5 times 4 times 3 times 2 times 1} 3003 )
Step 2: Calculate the Probability of Each Event
Event 1: Even Number of White Balls
The possible even counts of white balls when taking out 5 balls are 0, 2, and 4.
0 white balls, 5 black balls: ( {6 choose 0} cdot {9 choose 5} 1 cdot 126 126 ) 2 white balls, 3 black balls: ( {6 choose 2} cdot {9 choose 3} 15 cdot 84 1260 ) 4 white balls, 1 black ball: ( {6 choose 4} cdot {9 choose 1} 15 cdot 9 135 )Adding these values, we get:
Total for even white balls 126 1260 135 1521
Event 2: Exactly 3 Black Balls
This means we will have 2 white balls:
( {6 choose 2} cdot {9 choose 3} 15 cdot 84 1260 )
Event 3: Exactly 4 Black Balls
This means we will have 1 white ball:
( {6 choose 1} cdot {9 choose 4} 6 cdot 126 756 )
Step 3: Combine Events
Now we need to add the probabilities of these events, but we need to consider overlaps:
The case of 2 white balls and 3 black balls is already counted in the event for 3 black balls. The case of 1 white ball and 4 black balls is already counted in the event for 4 black balls.Thus we calculate the combined total as follows:
Total Count of Favorable Outcomes 1521 1260 756 - 1260 - 756 1521
Final Probability
The probability of the desired events is:
P ( frac{1521}{3003} )
This fraction simplifies to:
P ≈ 0.506 or 50.6%
Thus, the probability that we take out an even number of white balls or 3 black balls or 4 black balls is approximately 0.506 or 50.6%.
Conclusion
This step-by-step approach to calculating probability is not only mathematically sound but also serves as a useful example for SEO content writers. By providing detailed explanations of each step, the process becomes easier to understand and easier to reference for future problems. SEO optimization includes using relevant keywords, structuring content logically, and providing clear explanations, which are all evident in this article. The use of equations and explanations ensures that the content is both informative and accessible to a wide range of readers.