Problem 16 - 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 white given that the second ball drawn was white?

Step-by-Step Solution

Answer

The probability that the transferred ball was white given that the second ball drawn was white is \(\frac{8}{17}\).

Step 1: Define the events

Let W1 be the event of drawing a white ball from urn A and transferring it to urn B. Let W2 be the event of drawing a white ball from urn B after the transfer. We want to find the probability P(W1 | W2), which is the probability of event W1 happening given that event W2 has already happened.

Step 2: Calculate P(W1)

First, let's find the probability of drawing a white ball from urn A. There are a total of 10 balls in urn A, 4 of which are white. So, P(W1) = \(\frac{number\:of\:white\:balls\:in\:A}{total\:number\:of\:balls\:in\:A}\) = \(\frac{4}{10}\) = \(\frac{2}{5}\)

Step 3: Calculate P(W2 | W1) and P(W2 | ¬W1)

Now, we want to find the probability of drawing a white ball from urn B given that a white ball has been transferred from urn A. After the transfer, urn B would contain 4 white balls and 5 black balls (a total of 9 balls). So, P(W2 | W1) = \(\frac{number\:of\:white\:balls\:in\:B\:after\:transfer}{total\:number\:of\:balls\:in\:B\:after\:transfer}\) = \(\frac{4}{9}\) Next, we want to find the probability of drawing a white ball from urn B given that a black ball was transferred from urn A. In this case, urn B would contain 3 white balls and 6 black balls (a total of 9 balls). So, P(W2 | ¬W1) = \(\frac{number\:of\:white\:balls\:in\:B\:with\:black\:ball\:transfer}{total\:number\:of\:balls\:in\:B\:with\:black\:ball\:transfer}\) = \(\frac{3}{9}\) = \(\frac{1}{3}\)

Step 4: Use Bayes' Theorem to calculate P(W1 | W2)

Bayes' Theorem states that: \(P(W1|W2) = \frac{P(W2|W1) * P(W1)}{P(W2|W1) * P(W1) + P(W2|¬W1) * P(¬W1)}\) We already calculated P(W1), P(W2 | W1) and P(W2 | ¬W1). We also need to calculate P(¬W1), which is the probability of not drawing a white ball from urn A (in this case, drawing a black ball). P(¬W1) = 1 - P(W1) = 1 - \(\frac{2}{5}\) = \(\frac{3}{5}\) Now, we can plug in the values into Bayes' Theorem: \(P(W1|W2) = \frac{\frac{4}{9} * \frac{2}{5}}{\frac{4}{9} * \frac{2}{5} + \frac{1}{3} * \frac{3}{5}}\)

Step 5: Simplify and calculate the answer

Now, let's simplify the equation to get our final answer: \(P(W1|W2) = \frac{\frac{8}{45}}{\frac{8}{45} + \frac{9}{45}}\) = \(\frac{8}{45} * \frac{1}{\frac{17}{45}}\) = \(\frac{8}{17}\) So, the probability that the transferred ball was white given that the second ball drawn was white is \(\frac{8}{17}\).

Key Concepts

Bayes' Theorem

Understanding Bayes' Theorem is essential to solving conditional probability problems like the one in this exercise. Bayes' Theorem allows us to find the probability of an event, based on prior knowledge of conditions that might be related to the event. It is written mathematically as:\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]Here, \(P(A|B)\) represents the probability of event \(A\) occurring given that \(B\) has occurred.

• \(P(B|A)\) is the probability of \(B\) given \(A\).
• \(P(A)\) is the prior probability of event \(A\).
• \(P(B)\) is the total probability of event \(B\).

In the urn exercise, we use Bayes' Theorem to find \(P(W1|W2)\), where \(W1\) is the event of transferring a white ball from urn A, and \(W2\) is drawing a white ball from urn B after the transfer. By calculating these probabilities and applying them to the formula, Bayes' Theorem helps to unravel the nuanced relationship between events in probability theory.

Probability Theory

Probability Theory forms the foundation for understanding and solving problems involving uncertainty, like the probability issues presented in the urn exercise. This theory addresses the study of randomness and helps in determining the likelihood of different outcomes.In the exercise, we determine several crucial probabilities:

• \(P(W1)\): The chance of drawing a white ball from urn A.
• \(P(W2 | W1)\): The likelihood of picking a white ball from urn B given a white ball was transferred.
• \(P(W2 | eg W1)\): The probability of drawing a white ball from urn B given a black ball was transferred.

These probabilities help to piece together the elements needed for Bayes' Theorem. By converting routine situations (like drawing balls from urns) into probability expressions, Probability Theory aids in comprehending and computing the chance of various outcomes.

Mathematics Education

Mathematics Education strives to present students with the tools and frameworks necessary to tackle complex problems. By teaching concepts like Bayes' Theorem and Probability Theory, educators equip learners with the ability to interpret and analyze various scenarios involving chance and uncertainty. Concrete problems, such as the urn experiment, allow students to apply theoretical knowledge to practical situations, clarifying core mathematical principles. Methods such as:

• Defining events explicitly to understand problem statements.
• Breaking down the problem into smaller, more manageable steps.
• Learning to apply formulas and theorems correctly.

These pedagogical techniques ensure that learners not only solve the particular problem at hand but also gain a deeper understanding of underlying mathematical frameworks. When students see how concepts interconnect in real-world scenarios, they become better prepared for more complex mathematical challenges.