Introduction: Understanding Probability in Combinatorial Drawings
In this article, we will delve into a detailed step-by-step process to solve a probability problem involving a bag containing 12 black and 6 white balls. Specifically, we will determine the probability that in the next two draws exactly one white ball is drawn, given that at least 4 white balls were drawn in the initial 6 draws. We will use combinatorial analysis and probability laws to reach our conclusion.
Step 1: Total Balls and Initial Conditions
First, let's establish the total number of black and white balls. There are a total of 18 balls, composed of 12 black balls and 6 white balls.
Step 2: Drawing 6 Balls with at Least 4 White Balls
We need to consider three scenarios where at least 4 white balls are drawn in the initial 6 draws:
Scenario 1: Drawing 4 White and 2 Black Balls
The number of ways to choose 4 white balls from 6 and 2 black balls from 12:
Number of ways to choose 4 white balls from 6:
Number of ways to choose 2 black balls from 12:
Total combinations for this scenario:
Scenario 2: Drawing 5 White and 1 Black Ball
The number of ways to choose 5 white balls from 6 and 1 black ball from 12:
Number of ways to choose 5 white balls from 6:
Number of ways to choose 1 black ball from 12:
Total combinations for this scenario:
Scenario 3: Drawing 6 White and 0 Black Ball
The number of ways to choose 6 white balls from 6 and 0 black balls from 12:
Number of ways to choose 6 white balls from 6:
Number of ways to choose 0 black balls from 12:
Step 3: Total Combinations for At Least 4 White Balls
The total number of ways to draw at least 4 white balls is the sum of the combinations from the three scenarios.
Step 4: Total Ways to Draw Any 6 Balls
The total number of ways to draw any 6 balls from 18:
Step 5: Probability of Drawing At Least 4 White Balls
The probability of drawing at least 4 white balls is the ratio of the total combinations for at least 4 white balls to the total ways to draw any 6 balls.
Step 6: Next Two Draws
Now, let's consider the next two draws. We will analyze each case based on the number of white balls drawn in the first 6 draws.
Case 1: Drawing 4 White and 2 Black Balls
Remaining balls: 2 white, 10 black
Total combinations for exactly 1 white ball in the next 2 draws:
Probability for exactly 1 white ball in the next 2 draws:
Case 2: Drawing 5 White and 1 Black Ball
Remaining balls: 1 white, 11 black
Total combinations for exactly 1 white ball in the next 2 draws:
Probability for exactly 1 white ball in the next 2 draws:
Case 3: Drawing 6 White Balls
Remaining balls: 0 white, 12 black
No way to draw 1 white ball, so the probability is 0.
Step 7: Total Probability of Drawing Exactly 1 White Ball
We will use the law of total probability to combine the probabilities for each case.
Probability of each case and their respective probabilities for drawing exactly 1 white ball in the next 2 draws:
Combining the probabilities:
Final result:
The probability that in the next two draws exactly one white ball is drawn, given that at least 4 white balls were drawn initially is 20592/35079.
Conclusion
This problem demonstrates the application of combinatorial analysis and probability laws in solving complex drawing scenarios. Understanding these concepts provides a strong foundation for tackling similar probability problems in various fields, including data science, statistics, and gaming. Through a step-by-step approach, we can accurately calculate and interpret probabilities in such scenarios.