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Chapter 7: Problem 47

During 2008, approximately 2,750 million new listings were posted worldwide on eBay, of which 1,130 million were in the United States. A total of 730 million of the worldwide new listings, including 275 million in the United States, were posted during the fourth quarter of \(2008 .{ }^{.44}\) a. Find the probability that a new listing on eBay was in the United States, given that the item was posted during the fourth quarter. b. Find the probability that a new listing on eBay was posted in the fourth quarter, given that the item was posted in the United States.

Short Answer

Expert verified
a. The probability that a new listing on eBay was in the United States, given that the item was posted during the fourth quarter is approximately \(0.3767\) or \(37.67\%\). b. The probability that a new listing on eBay was posted in the fourth quarter, given that the item was posted in the United States, is approximately \(0.2434\) or \(24.34\%\).

Step by step solution

01

a. Probability of the new listing being in the United States, given that it was posted during the fourth quarter.

To find this probability, we can use the conditional probability formula. Here, A represents the event that the new listing is in the United States, and B represents the event that the new listing was posted during the fourth quarter. We need to find P(A|B): First, find the probability intersection P(A 鈭� B) which represents the probability that a new listing is in the United States and posted during the fourth quarter: \[P(A \cap B) = \frac{\text{US new listings in fourth quarter}}{\text{Worldwide new listings}}\] \[P(A \cap B) = \frac{275\text{ million}}{2750\text{ million}}\] Next, find the probability of P(B), which represents the probability that a new listing is posted during the fourth quarter: \[P(B) = \frac{\text{Worldwide new listings in fourth quarter}}{\text{Worldwide new listings}}\] \[P(B) = \frac{730\text{ million}}{2750\text{ million}}\] Now, we can apply the conditional probability formula: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\] \[P(A|B) = \frac{\frac{275\text{ million}}{2750\text{ million}}}{\frac{730\text{ million}}{2750\text{ million}}}\] Finally, calculate the probability: \[P(A|B) \approx 0.3767\] The probability that a new listing was in the United States, given that it was posted during the fourth quarter, is approximately 0.3767 (37.67%).
02

b. Probability of the new listing being posted in the fourth quarter, given that it was in the United States.

Now, we will find the probability P(B|A), the probability that a new listing was posted in the fourth quarter, given that it was posted in the United States: First, find the probability P(A), which represents the probability that a new listing is in the United States: \[P(A) = \frac{\text{US new listings}}{\text{Worldwide new listings}}\] \[P(A) = \frac{1130\text{ million}}{2750\text{ million}}\] Now, apply the conditional probability formula: \[P(B|A) = \frac{P(A \cap B)}{P(A)}\] \[P(B|A) = \frac{\frac{275\text{ million}}{2750\text{ million}}}{\frac{1130\text{ million}}{2750\text{ million}}}\] Finally, calculate the probability: \[P(B|A) \approx 0.2434\] The probability that a new listing was posted in the fourth quarter, given that it was posted in the United States, is approximately 0.2434 (24.34%).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Probability Theory
Probability theory is the mathematical study of uncertainty and events. It is a fundamental aspect of statistics and is used to describe the likelihood of specific outcomes occurring. When we refer to the 'probability' of an event, we are discussing a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

The probability of an event can be calculated by taking the number of favorable outcomes and dividing it by the total number of possible outcomes. In the context of the eBay listings discussed in the exercise, probability theory helps us predict the likelihood of where and when listings will be posted based on previous data.

For instance, when we are looking for the probability that a listing was from the United States, we consider the total number of US listings as our favorable outcomes and all listings worldwide as the possible outcomes. To make probability theory more practical and understandable, consider concepts like sample space (total possible outcomes), events (specific outcomes of interest), and the idea that the sum of probabilities of all possible outcomes must equal 1.
Demystifying Bayes' Theorem
Bayes' Theorem is a powerful equation in probability theory that allows us to update our predictions or beliefs in light of new evidence. It relates the conditional probability of an event, given another, to the inverse condition. Simply put, it tells us how the likelihood of an event changes with new information.

Bayes' Theorem can be stated as follows: \[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\]
where:
    \t
  • \t\(P(A|B)\) - the probability of event A occurring given that B has occurred.\t
    • \t
  • \t\(P(B|A)\) - the probability of event B occurring given that A has occurred.\t
    • \t
  • \t\(P(A)\) - the probability of event A occurring on its own.\t
    • \t
  • \t\(P(B)\) - the probability of event B occurring on its own.\t
In the eBay exercise, Bayes' Theorem would help us update our beliefs about the probability of a listing being in the fourth quarter, given that we know it is from the United States or vice versa. This theorem is instrumental when navigating through complexities of dependent or conditional probabilities.
Event Intersection in Probability
Event intersection refers to a situation where two events are both considered at the same time. In probability, the intersection of events A and B, denoted as \(A \cap B\), is the event that both A and B occur.

When we calculate the probability of the intersection of two events, we are seeking the likelihood that both events happen together. The formula for the intersection of two events, assuming they are not mutually exclusive, is: \[P(A \cap B) = P(A) \cdot P(B|A)\]
This formula states that the probability of both events occurring is equal to the probability of event A occurring multiplied by the probability that event B occurs given that event A has occurred.

In our eBay example, the event intersection is used to find the probability of listings that are both in the United States and posted during the fourth quarter. Understanding intersections is essential when events are not independent, meaning the occurrence of one affects the likelihood of the other. This concept is necessary for calculating conditional probabilities and is utilized in both parts of the eBay exercise.

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