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Chapter 4: Problem 41

A consulting firm submitted a bid for a large research project. The firm's management initially felt they had a \(50-50\) chance of getting the project. However, the agency to which the bid was submitted subsequently requested additional information on the bid. Past experience indicates that for \(75 \%\) of the successful bids and \(40 \%\) of the unsuccessful bids, the agency requested additional information. a. What is the prior probability of the bid being successful (that is, prior to the request for additional information)? b. What is the conditional probability of a request for additional information given that the bid will ultimately be successful? c. Compute the posterior probability that the bid will be successful given a request for additional information.

Short Answer

Expert verified
a) 0.5 b) 0.75 c) 0.6522

Step by step solution

01

Determine the Prior Probability

The prior probability is the initial belief about the event of the bid being successful. According to the problem, the firm's management initially felt they had a 50-50 chance of getting the project. Therefore, the prior probability of the bid being successful is:\[P(S) = 0.5\]
02

Conditional Probability of Request Given Success

The conditional probability that additional information is requested given that the bid is successful is provided in the problem statement. It is stated that for 75% of the successful bids, the agency requested additional information. Therefore:\[P(R|S) = 0.75\]where \( R \) represents the request for additional information and \( S \) represents a successful bid.
03

Determine Additional Probability Components

We are also given that for 40% of unsuccessful bids, the agency requests additional information. Therefore, the probability of a request given an unsuccessful bid is:\[P(R|U) = 0.4\]The prior probability of the bid being unsuccessful, \( U \), is complementary to the probability of the bid being successful:\[P(U) = 1 - P(S) = 0.5\]
04

Compute Posterior Probability Using Bayes' Theorem

To find the posterior probability that the bid is successful given a request for additional information, we use Bayes' theorem:\[P(S|R) = \frac{P(R|S) \cdot P(S)}{P(R)}\]First, calculate \( P(R) \), the total probability of a request:\[P(R) = P(R|S) \cdot P(S) + P(R|U) \cdot P(U) = 0.75 \cdot 0.5 + 0.4 \cdot 0.5 = 0.375 + 0.2 = 0.575\]Now substitute these into Bayes' theorem:\[P(S|R) = \frac{0.75 \cdot 0.5}{0.575} = \frac{0.375}{0.575} \approx 0.6522\]

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Key Concepts

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

Probability Theory
Probability theory is a mathematical framework that assists us in assessing the likelihood of various outcomes of uncertain events. When dealing with probability theory, we initially begin by defining the basic components.
- **Outcome**: The result of a single execution or trial of an experiment.
- **Event**: One or more outcomes collectively forming a specific occurrence. For instance, rolling a die and landing on either a 2 or a 4 can be described as an event.
- **Probability**: A numerical measure of the likelihood that an event occurs, commonly represented as a value between 0 and 1. A probability of 0 means an event will not happen, while a probability of 1 suggests certainty. The core principle includes the concept of sum of probabilities, where the total probability of all possible outcomes of a trial must equal 1. In the context of probability theory, Bayes' Theorem is a critical rule that parallels this principle by giving us a formal methodology to update probabilities with new information.
Conditional Probability
Conditional probability revolves around assessing the probability of an event occurring, given that another event has already taken place. In other words, it describes how our uncertainty about the outcome of an event changes with new information.
- The conditional probability of an event A given that another event B has occurred is denoted as \( P(A|B) \).
- It is calculated by the formula: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), provided \( P(B) > 0 \).Understanding the concept of conditional probability is important, as it allows us to refine our predictions. In our exercise, the probability of requesting additional information given a successful bid is a direct example. Expressed in values, this translates to \( P(R|S) = 0.75 \), reflecting the impact of the event of the bid being successful on the request.
Posterior Probability
Posterior probability represents an updated probability of an event after taking into consideration new evidence or information. This concept is central in applications of Bayes' theorem.
- Bayes' theorem allows us to calculate posterior probability by combining prior probability with new conditional probability.
- The formula is \( P(S|R) = \frac{P(R|S) \cdot P(S)}{P(R)} \), where \( P(S|R) \) is the probability of a bid being successful given a request for more information.In the exercise, after calculating other probabilities, such as the total probability of receiving a request \( P(R) \), we find the posterior probability that a bid is successful given the request, which is approximately \( 0.6522 \). Our updated belief considers both the initial estimate and the new data regarding the additional request. Thus, posterior probability helps refine predictions, offering a more informed estimate.

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Most popular questions from this chapter

An experiment has four equally likely outcomes: \(E_{1}, E_{2}, E_{3},\) and \(E_{4}\). a. What is the probability that \(E_{2}\) occurs? b. What is the probability that any two of the outcomes occur (e.g., \(E_{1}\) or \(E_{3}\) )? c. What is the probability that any three of the outcomes occur (e.g., \(E_{1}\) or \(E_{2}\) or \(E_{4}\) )?

An experiment with three outcomes has been repeated 50 times, and it was learned that \(E_{1}\) occurred 20 times, \(E_{2}\) occurred 13 times, and \(E_{3}\) occurred 17 times. Assign probabilities to the outcomes. What method did you use?

An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities. \\[ \begin{aligned} P(\text { high-quality oil }) &=.50 \\ P(\text { medium-quality oil }) &=.20 \\ P(\text { no oil }) &=.30 \end{aligned} \\] a. What is the probability of finding oil? b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test follow. \\[ \begin{aligned} P(\text { soil } | \text { high-quality oil }) &=.20 \\ P(\text { soil } | \text { medium-quality oil }) &=.80 \\ P(\text { soil } | \text { no oil }) &=.20 \end{aligned} \\] How should the firm interpret the soil test? What are the revised probabilities, and what is the new probability of finding oil?

Simple random sampling uses a sample of size \(n\) from a population of size \(N\) to obtain data that can be used to make inferences about the characteristics of a population. Suppose that, from a population of 50 bank accounts, we want to take a random sample of 4 accounts in order to learn about the population. How many different random samples of 4 accounts are possible?

Do you think the government protects investors adequately? This question was part of an online survey of investors under age 65 living in the United States and Great Britain (Financial Times/Harris Poll, October 1,2009 ). The number of investors from the United States and the number of investors from Great Britain who answered Yes, No, or Unsure to this question are provided as follows. a. Estimate the probability that an investor living in the United States thinks the government is not protecting investors adequately. b. Estimate the probability that an investor living in Great Britain thinks the government is not protecting investors adequately or is unsure the government is protecting investors adequately. c. For a randomly selected investor from these two countries, estimate the probability that the investor thinks the government is not protecting investors adequately. d. Based upon the survey results, does there appear to be much difference between the perceptions of investors living in the United States and investors living in Great Britain regarding the issue of the government protecting investors adequately?
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