Question

Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in an urn 2\. A ball is then drawn from urn 2. It happens to be white. What is the probability that the transferred ball was white?

Short answer

The probability that the transferred ball was white, given that a white ball was drawn from urn 2, is \( \frac{4}{5} \).

Step-by-step solution

Define the Events

Let's define the following events: - Event A: Transferring a white ball from urn 1 to urn 2 - Event B: Transferring a black ball from urn 1 to urn 2 - Event C: Drawing a white ball from urn 2 We want to find the probability of event A, given that event C has occurred (i.e., P(A|C)).

Calculate the Initial Probabilities

First, let's find the probabilities of events A and B, which represent the probability of transferring a white or black ball from urn 1, respectively. In urn 1, there are two white balls and one black ball. - P(A) = Probability of transferring a white ball from urn 1 = \( \frac{2}{3} \) - P(B) = Probability of transferring a black ball from urn 1 = \( \frac{1}{3} \)

Calculate the Probability of Drawing a White Ball from Urn 2

Now, we want to find the probabilities of drawing a white ball from urn 2, conditioned on events A and B. 1. If event A occurs (transferring a white ball from urn 1 to urn 2): - Urn 2 will then have two white balls and five black balls. - P(C|A) = Probability of drawing a white ball from urn 2, given that event A has occurred = \( \frac{2}{7} \) 2. If event B occurs (transferring a black ball from urn 1 to urn 2): - Urn 2 will then have one white ball and six black balls. - P(C|B) = Probability of drawing a white ball from urn 2, given that event B has occurred = \( \frac{1}{7} \)

Use Bayes' Theorem to Find the Desired Probability

We want to find P(A|C), i.e., the probability that the transferred ball was white, given that a white ball was drawn from urn 2. We can use Bayes' theorem for this purpose: P(A|C) = \( \frac{P(C|A) * P(A)}{P(C|A) * P(A) + P(C|B) * P(B)} \) Now substitute the values we found in the previous steps: P(A|C) = \( \frac{(\frac{2}{7}) * (\frac{2}{3})}{(\frac{2}{7}) * (\frac{2}{3}) + (\frac{1}{7}) * (\frac{1}{3})} \)

Simplify and Find the Final Probability

Finally, simplify the expression to find the desired probability: P(A|C) = \( \frac{\frac{4}{21}}{\frac{4}{21} + \frac{1}{21}} \) = \( \frac{4}{4 + 1} \) = \( \frac{4}{5} \) The probability that the transferred ball was white, given that a white ball was drawn from urn 2, is \( \frac{4}{5} \).

Key concepts

Bayes' Theorem

Bayes' Theorem is a powerful formula used in probability theory to update the probability estimate for an event based on new evidence. It addresses the question of computing the conditional probability of a hypothesis given observed data.

Using Bayes' Theorem, we can reverse the conditional probabilities. For instance, if we're interested in finding the likelihood of a cause given an observed effect, this theorem is invaluable. It is an integral part of a broader topic known as Bayesian inference, which is useful across various fields including statistics, medicine, and machine learning.

In our urn example, we have a scenario with two conditional events. We know that Event C (drawing a white ball from urn 2) has occurred, and we want to find the probability of Event A (transferring a white ball from urn 1 to urn 2) given this knowledge. In this context, Bayes' Theorem helps us to unravel this dependency by using the probabilities of the individual events and how they relate to one another.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena and modeling the likelihood of outcomes. It is fundamental to a wide variety of fields, including finance, science, engineering, and more.

In terms of our urn exercise, we apply probability theory to predict the likelihood of drawing a particular ball from a given set. The calculations start with the determination of initial probabilities, counting the favorable outcomes over the total number of outcomes, which is a basic principle in probability theory known as the classical definition of probability.

Understanding Event A and Event B, transferring a white or black ball respectively, and calculating the resultant probabilities of drawing a white ball from urn 2 (Event C) are quintessential exercises in probability theory. We use combinatorics to count possible outcomes and conditional probability to handle dependent events.

Combinatorics

Combinatorics is a field of mathematics focused on counting, and is often used in the study of probability theory. It involves calculating the number of ways certain outcomes can occur, which is essential for assessing probabilities. In the context of our urn problem, we use combinatorics to count the different ways in which balls can be drawn and transferred between urns.

In the initial setup, combinatorics helps us to understand that there are three possible balls to choose from urn 1, paving the way to establish the primary event probabilities, P(A) and P(B). This counting informs the foundation of our probability calculations and leads to determining the likeliness of the subsequent event of drawing a white ball from urn 2. By effectively applying the principles of combinatorics, we are able to simplify complex probabilistic scenarios into manageable equations that yield concrete probabilities.