Conditional Probability and Bayes' Theorem: Solving the White Ball Transfer Problem

Conditional probability is a fundamental concept in probability theory, and its application can be seen in a range of real-world scenarios. One such scenario involves a vessel containing 3 white and 5 black balls from which 4 balls are transferred to an empty vessel. If a white ball is drawn from this vessel, the question arises: what is the probability that exactly 3 out of the 4 transferred balls were white? This problem can be approached using Bayes' Theorem to determine the conditional probability. In this article, we will explore how to solve this problem step-by-step.

Introduction to the Problem

In a vessel, we have 3 white balls and 5 black balls. We transfer 4 balls to an empty vessel. Given that a white ball is drawn from this new vessel, we need to find the probability that 3 out of these 4 balls were white. This scenario can be mathematically modeled using conditional probability and Bayes' Theorem.

Theoretical Foundations

Bayes' Theorem

Bayes' Theorem is a formula for finding conditional probability. It is stated as follows:

P(A|B) (P(B|A) * P(A)) / P(B)

Event Definitions

In this problem, we define the following events:

A: The event that 3 white balls and 1 black ball are transferred. B: The event that a white ball is drawn from the second vessel.

We need to find P(A|B), the probability of event A given that event B has occurred.

Step-by-Step Solution

Step 1: Calculate P(A)

The probability of transferring 3 white and 1 black ball (event A) can be calculated using combinations. The total ways to choose 4 balls from 8 is:

P(A) (Ways to choose 3 white from 3) * (Ways to choose 1 black from 5) / (Total ways to choose 4 balls from 8)

Calculating the combinations:

Ways to choose 3 white from 3: (binom{3}{3} 1) Ways to choose 1 black from 5: (binom{5}{1} 5) Total ways to choose 4 balls from 8: (binom{8}{4} 70)

Therefore, P(A) (1 * 5) / 70 1/14.

Step 2: Calculate P(B|A)

Given that 3 white and 1 black ball are transferred, the probability of drawing a white ball (event B given A) is:

P(B|A) Number of white balls / Total balls 3/4.

Step 3: Calculate P(B)

Next, we need to calculate the total probability of drawing a white ball (event B). We consider all possible combinations of balls that could be transferred:

Case 1: 3 white and 1 black (A): P(A) * P(B|A) (1/14) * (3/4) 3/56 Case 2: 2 white and 2 black (2W 2B): P(2W 2B) * P(B|2W 2B) (3/7) * (1/2) 3/14 12/56 Case 3: 1 white and 3 black (1W 3B): P(1W 3B) * P(B|1W 3B) (3/7) * (1/4) 3/28 6/56 Case 4: 0 white and 4 black (0W 4B): P(0W 4B) * P(B|0W 4B) 0 * 0 0

Now, we can calculate P(B):

P(B) P(A) * P(B|A) P(2W 2B) * P(B|2W 2B) P(1W 3B) * P(B|1W 3B) P(0W 4B) * P(B|0W 4B)

Substituting the values:

P(B) (3/56) (12/56) (6/56) 0 21/56 3/8.

Step 4: Calculate P(A|B)

Finally, we substitute back into Bayes' Theorem to find P(A|B):

P(A|B) (P(B|A) * P(A)) / P(B) (3/4 * 1/14) / (3/8) (3/56) / (3/8) (3/56) * (8/3) 8/56 1/7.

Final Answer: The probability that out of the four balls transferred, 3 are white and 1 is black, given that a white ball is drawn, is (boxed{1/7}).

Conclusion

This problem demonstrates the power of Bayes' Theorem in solving conditional probability problems. By breaking down the problem into manageable steps and applying the fundamental principles of probability and combinations, we can arrive at the correct solution. This approach is not only useful for solving such theoretical problems but also has practical applications in fields such as statistics, data science, and artificial intelligence.