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Chapter 4: Problem 55

Suppose a group of research proposals was evaluated by a panel of experts to decide whether or not they were worthy of funding. When these same proposals were submitted to a second independent panel of experts, the decision to fund was reversed in \(30 \%\) of the cases. If the probability that a proposal is judged worthy of funding by the first panel is \(.2,\) what are the probabilities of these events? a. A worthy proposal is approved by both panels. b. A worthy proposal is disapproved by both panels. c. A worthy proposal is approved by one panel.

Short Answer

Expert verified
To summarize, for a worthy research proposal: a. The probability of being approved by both panels is 0.14. b. The probability of being disapproved by both panels is 0.24. c. The probability of being approved by one panel is 0.62.

Step by step solution

01

a. A worthy proposal is approved by both panels.

We want the probability of \(P(A1 \cap A2 | A1)\). Since we know \(P(A1) = 0.2\), and the second panel reverses the decision in \(30\%\) of the cases, we can say \(P(A2 | A1) = 1 - 0.3 = 0.7\). Thus, we have \(P(A1 \cap A2 | A1) = P(A2 | A1) \cdot P(A1) = 0.7 \cdot 0.2 = 0.14\).
02

b. A worthy proposal is disapproved by both panels.

In this case, we want the probability of \(P(D1 \cap D2 | D1)\). Since we have \(P(D1) = 1 - P(A1) = 1 - 0.2 = 0.8\), and the second panel reverses the decision in \(30\%\) of the cases, we can say \(P(D2 | D1) = 0.3\). Thus, we get \(P(D1 \cap D2 | D1) = P(D2 | D1) \cdot P(D1) = 0.3 \cdot 0.8 = 0.24\).
03

c. A worthy proposal is approved by one panel.

For this case, we want the probability of \(P((A1 \cap D2) \cup (D1 \cap A2))\). Using the information from the earlier cases, we have: \(P(A1 \cap D2) = P(D2 | A1) \cdot P(A1) = 0.3 \cdot 0.2 = 0.06\). \(P(D1 \cap A2) = P(A2 | D1) \cdot P(D1) = 0.7 \cdot 0.8 = 0.56\). Now, we can find the probability of both events: \(P((A1 \cap D2) \cup (D1 \cap A2)) = P(A1 \cap D2) + P(D1 \cap A2) = 0.06 + 0.56 = 0.62\). So, we have found the probabilities of the following events for a worthy research proposal: a. Approved by both panels: \(0.14\) b. Disapproved by both panels: \(0.24\) c. Approved by one panel: \(0.62\)

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Key Concepts

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

Independent Events
In probability theory, independent events are those whose occurrence or outcome does not affect each other. This means that knowledge of one event tells you nothing about the other. For example, when flipping a coin, each coin flip is independent of others. Understanding independent events can help simplify complex probability scenarios.
  • If two events, A and B, are independent, the probability of both happening is the product of their individual probabilities: \(P(A \cap B) = P(A) \cdot P(B)\).
  • In the context of our exercise, the decision by one panel does not directly influence the decision of another, assuming they're independent. This means that the outcome of one panel remains independent unless stated otherwise.
For instance, if you want to realize if the panels' decisions are truly independent, you'll need more than just the given probabilities; you would need evidence showing that the choice of one panel has no influence on the other.
Conditional Probability
Conditional probability refers to the probability of an event occurring given that another event has already occurred. This is denoted as \(P(A|B)\), which represents the probability of event A happening given that B has occurred. It is a crucial concept in scenarios where one event potentially affects another.
  • The formula for conditional probability is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), where \(P(A \cap B)\) is the joint probability of both events A and B occurring simultaneously.
  • In our exercise, when considering the probability that a worthy proposal is approved by both panels, we are looking at a conditional setup. Mainly, you're determined the probability of a second panel's approval given the first one's approval; this is \(P(A2 | A1)\), informing us that if the first panel approves, there's a 70% chance the second one will too.
By accurately utilizing conditional probability, you can better estimate outcomes in dependent scenarios.
Joint Probability
Joint probability is concerned with the likelihood of two events happening at the same time. It is written as \(P(A \cap B)\). Joint probability is different from conditional probability because it does not rely on having prior information about one of the events.
  • The calculation method involves taking the product of the probability of both events occurring together, provided they are independent, or using known conditional probabilities otherwise.
  • In the exercise example, for a proposal to be approved by both panels, you needed to find the joint probability. This utilizes both the conditional probability of the second event given the first and the independent probability of the first event: \(P(A1 \cap A2) = P(A2 | A1) \cdot P(A1)\).
Understanding joint probability enables you to predict the likelihood of multiple events happening in tandem, critical especially in planning and making decisions that are contingent on multiple factors.

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Most popular questions from this chapter

Three people are randomly selected to report for jury duty. The gender of each person is noted by the county clerk. a. Define the experiment. b. List the simple events in \(S\). c. If each person is just as likely to be a man as a woman, what probability do you assign to each simple event? d. What is the probability that only one of the three is a man? e. What is the probability that all three are women?

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