Problem 23 - A First Course in Probability

Question

Urn I contains 2 white and 4 red balls, whereas urn II contains 1 white and 1 red ball. A ball is randomly chosen from urn I and put into urn II, and a ball is then randomly selected from urn II. What is (a) the probability that the ball selected from urn II is white? (b) the conditional probability that the transferred ball was white given that a white ball is selected from urn II?

Step-by-Step Solution

Answer

(a) The probability that the ball selected from urn II is white is \(\frac{4}{9}\). (b) The conditional probability that the transferred ball was white given that a white ball is selected from urn II is \(\frac{1}{2}\).

Step 1: List down the possibilities and probabilities of ball transfers

First, let's identify the possible scenarios when transferring a ball from urn I to urn II. 1. White ball is transferred from urn I to urn II (W1): The probability of this event is \(P(W1) = \frac{2}{6} = \frac{1}{3}\) 2. Red ball is transferred from urn I to urn II (R1): The probability of this event is \(P(R1) = \frac{4}{6} = \frac{2}{3}\) Now let's list the possibilities when a ball is later randomly selected from urn II: 1. White ball is selected from urn II (W2): We will calculate its probability later (i.e., in step 3) through conditional probability. 2. Red ball is selected from urn II (R2): We will calculate its probability later (i.e., in step 3) through conditional probability.

Step 2: Identify the relationship between the different scenarios

We will denote the probability of selecting a white ball from urn II (W2) given that a white ball was transferred from urn I (W1) by P(W2 | W1). Similarly, we will denote the probability of selecting a white ball from urn II (W2) given that a red ball was transferred from urn I (R1) by P(W2 | R1).

Step 3: Calculate the conditional probabilities P(W2 | W1) and P(W2 | R1)

We can now calculate the following conditional probabilities: 1. The probability of selecting a white ball from urn II given that a white ball was transferred from urn I: \(P(W2 | W1) = \frac{2}{3}\) (because now urn II contains 2 white balls and 1 red ball) 2. The probability of selecting a white ball from urn II given that a red ball was transferred from urn I: \(P(W2 | R1) = \frac{1}{3}\) (because now urn II contains 1 white ball and 2 red balls)

Step 4: Calculate the probability that the ball selected from urn II is white (Part a of the problem)

To find the probability that the ball selected from urn II is white, we can use the law of total probability. Let's apply the formula and plug in the probabilities we found earlier: \(P(W2) = P(W2 | W1) \times P(W1) + P(W2 | R1) \times P(R1) = \frac{2}{3} \times \frac{1}{3} + \frac{1}{3} \times \frac{2}{3} = \frac{2}{9} + \frac{2}{9} = \frac{4}{9}\) So, the probability that the ball selected from urn II is white is \(\frac{4}{9}\).

Step 5: Calculate the conditional probability that the transferred ball was white given that a white ball is selected from urn II (Part b of the problem)

We are now asked to find the conditional probability that the transferred ball was white given that a white ball is selected from urn II, which can be denoted as \(P(W1 | W2)\). We can find this by using Bayes' theorem: \(P(W1 | W2) = \frac{P(W2 | W1) \times P(W1)}{P(W2)} = \frac{\frac{2}{3} \times \frac{1}{3}}{\frac{4}{9}} = \frac{\frac{2}{9}}{\frac{4}{9}} = \frac{1}{2}\) So, the conditional probability that the transferred ball was white given that a white ball is selected from urn II is \(\frac{1}{2}\).

Key Concepts

Law of Total Probability

The Law of Total Probability is a fundamental rule used to find probabilities of various outcomes by breaking them down into simpler events. It's especially useful in situations where multiple ways can lead to a particular outcome. Understanding this law helps us to see the whole picture by considering smaller, separate events. For the urn problem, the law helps us calculate the probability of selecting a white ball from urn II by considering both scenarios of transferring a ball from urn I—either a white ball or a red.

Here's the idea broken down:

• Identify all possible ways an event can happen—in our case, both transferring a white or a red ball from urn I.
• Calculate the probability of each of these easier events happening.
• Add up the probabilities of these paths, weighted by their chance of happening overall, to get the total probability of our desired event—selecting a white ball from urn II.

Using the law of total probability, we combined the separate conditional probabilities to find that the probability of picking white from urn II is \(\frac{4}{9}\). This powerful tool translates complex events into a series of simpler, intuitive probabilities.

Bayes' Theorem

Bayes' Theorem is a remarkable formula in conditional probability that allows us to update probabilities with new information. This theorem helps us determine the likelihood of an earlier event, given that a later event has occurred. It's a bridge that connects conditional probabilities in a systematic way. In the urn example, Bayes' theorem helps us solve the second part of the exercise: finding the probability that the transferred ball was white, given that a white ball was eventually drawn from urn II.

This theorem uses the relationship:

• The probability of the later event happening (selecting a white ball).
• The probability of both events occurring together (transferring and selecting a white ball).
• The probability of the first event happening on its own (transferring a white ball).

When we applied Bayes' theorem, it showed us that the probability of having transferred a white ball when a white one was picked from urn II is \(\frac{1}{2}\). This theorem is all about updating our understanding as new evidence comes in, forming a keystone in probability calculations.

Probability Calculation

Probability calculations involve determining how likely an event is to occur. They form the foundation of predictions and decisions in uncertain circumstances. Understanding these calculations involves grasping several key principles, such as understanding probability as a ratio of favorable outcomes to the total number of outcomes. Let’s delve deeper into how this applies to the urn exercise.

When calculating the probability of extracting a white ball from an urn:

• Determine the total number of possible outcomes—how many balls you could pick.
• Count the number of favorable outcomes—balls that meet your criteria (white balls in our example).
• Formulate the probability as a fraction: the number of favorable outcomes over the total outcomes.

In our scenario, transferring balls affects these counts, thus affecting the probabilities. For example, when a white ball is moved to urn II, the count of whites and reds changes, influencing the probability of selecting a white one. Every change in circumstances alters the likelihood, making it crucial to accurately count and apply these calculations.

Urn Problems

Urn problems are classic exercises in probability theory, providing a structured way to understand complex events. They involve selecting items from containers (in our case, balls from an urn), and they come in many variations. These problems help in visualizing conditional and combined probabilities with straightforward setups.

Here's what makes urn problems so helpful:

• They offer a concrete model for thinking about probabilities and random variables.
• They help illustrate concepts like independence and conditional probability.
• They act as practical illustrations of complex theorems, including the Law of Total Probability and Bayes' Theorem.

In this specific problem, two urns with different compositions of colored balls show how transferring and drawing affect outcomes. By altering the contents of one urn and examining the result, we explore how different conditional probabilities play out. Urn problems are useful because they transform abstract math into tangible, real-world situations, providing a solid base for further exploration in probability.