Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to the following experiment: Urn A contains four white and six black balls. Urn B contains three white and five black balls. A ball is drawn from urn A and then transferred to urn B. A ball is then drawn from urn B. What is the probability that the transferred ball was black given that the second ball drawn was white?

Step-by-Step Solution

Answer

The probability that the transferred ball was black given that the second ball drawn from urn B was white is \(P(A|B) = \frac{12}{17}\).

Step 1: Define Events

Let's define the two events as follows: - Event A: Transferred ball is black - Event B: Ball drawn from urn B is white We need to find the probability P(A|B), which is the probability that the transferred ball was black given that the second ball drawn from urn B was white.

Step 2: Conditional Probability Formula

The conditional probability formula is given by: P(A|B) = \(\frac{P(A \cap B)}{P(B)}\) Where: - P(A|B) is the probability of A happening given B happened - P(A ∩ B) is the probability of both A and B happening - P(B) is the probability of B happening

Step 3: Calculate P(A ∩ B) - probability of both events happening

To find the probability of both events happening, we'll first calculate the probability of drawing a black ball from urn A and then transferring it to urn B, and then drawing a white ball from urn B (now containing 4 white and 6 black balls). This can be calculated as follows: P(A ∩ B) = P(A) * P(B|A) = \(\frac{6}{10}\) * \(\frac{4}{9}\)= \(\frac{24}{90}\)

Step 4: Calculate P(B) - probability of drawing a white ball from urn B

We can find the probability of drawing a white ball from urn B after transferring a ball from urn A by considering both possibilities: 1. If a white ball is transferred from urn A to urn B, the probability of drawing a white ball from urn B is: \(\frac{4}{10}\) * \(\frac{4}{9}\) = \(\frac{16}{90}\) 2. If a black ball is transferred from urn A to urn B, the probability of drawing a white ball from urn B is: \(\frac{6}{10}\) * \(\frac{3}{9}\) = \(\frac{18}{90}\) Now, we can calculate the total probability of drawing a white ball from urn B: P(B) = \(\frac{16}{90}\) + \(\frac{18}{90}\) = \(\frac{34}{90}\)

Step 5: Apply the Conditional Probability Formula

Now, using the conditional probability formula, we can find the probability that the transferred ball was black given that the second ball drawn was white: P(A|B) = \(\frac{P(A \cap B)}{P(B)}\) = \(\frac{\frac{24}{90}}{\frac{34}{90}}\) = \(\frac{24}{34}\)

Step 6: Simplifying the Result

Finally, we need to simplify the result. The fraction \(\frac{24}{34}\) can be simplified by dividing both the numerator and the denominator by 2: P(A|B) = \(\frac{12}{17}\) This is the final answer, and it represents the probability that the transferred ball was black given that the second ball drawn from urn B was white.

Key Concepts

Understanding Probability Theory

Probability theory is the mathematical framework that allows us to analyze chance and uncertainty. It provides the tools to determine the likelihood of various outcomes in an experiment or process. A key foundation of probability theory is the concept of an 'event', which refers to a specific outcome or a combination thereof. For instance, when flipping a coin, the possible events are 'heads' or 'tails'.

There are several types of probabilities, including marginal, joint, and conditional probabilities. Marginal probability refers to the likelihood of a single event occurring in isolation. Joint probability is concerned with the likelihood of two or more events happening simultaneously. Conditional probability, the main focus in our problem scenario, looks at the likelihood of an event occurring given that another event has already happened.

Understanding these different types of probabilities and how they interrelate is crucial for problems in a wide array of fields, from simple gambling games to complex statistical analysis and decision-making.

Applying Bayes' Theorem

Bayes' theorem is a fundamental result of probability theory that allows one to update the probability estimate for an event, as more information or evidence becomes available. It relates the conditional probabilities of two events in a reciprocal manner.

Bayes' theorem is mathematically expressed as:
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]
Here, \( P(A|B) \) is the probability of event A occurring given that B is true. Conversely, \( P(B|A) \) is the probability of event B given that A is true. \( P(A) \) and \( P(B) \) are the probabilities of A and B occurring independently. Bayes’ theorem is instrumental in many fields, including statistics, economics, medicine, and more—wherever the revision of predictions or hypotheses in the light of new evidence is necessary.

In the given exercise, we apply Bayes' theorem to find the likelihood that the transferred ball is black given that a white ball has been drawn from urn B. Understanding this theorem is crucial for solving these types of conditional probability problems.

The Role of Combinatorics

Combinatorics is a branch of mathematics dealing with the counting, combination, and arrangement of set elements. In probability theory, combinatorics helps quantify the number of ways that certain outcomes can occur, which is essential for calculating probabilities in more complex scenarios, precisely the types of scenarios encountered when dealing with multiple events or elements.

Two principal combinatorial concepts found in probability problems are permutations and combinations. Permutations consider the arrangement of objects where the order is important, while combinations consider groups of items where the order does not matter. Our problem doesn't explicitly utilize intricate combinatorial calculations, but understanding how the balls are selected in sequence—a fundamental combinatorial action—is intrinsic to the solution.

In probability exercises, combinatorics allows us to calculate the total number of possible outcomes, which is often required in the denominator of probability fractions. Thus, a strong grasp of combinatorics underpins a better understanding of probability theory and, ultimately, guides us to correct solutions in exercises such as the transfer of balls between urns.