Question

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 that: 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

Question: Calculate the probability for the following scenarios: (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. Answer: Using conditional probabilities and the given probabilities, the probabilities for the scenarios are: (a) 0.14, (b) 0.06, and (c) 0.2.

Step-by-step solution

Identify the given probabilities

We are given: 1. The probability that a proposal is judged worthy of funding by the first panel, \(P(W_1) = 0.2\). 2. The probability that the funding decision is reversed by the second panel, \(P(R) = 0.3\).

Calculate the probabilities of proposals being worthy or unworthy of funding in panel 1

The probability that a proposal is judged unworthy of funding by the first panel is given by the complement of \(P(W_1)\). Thus, \(P(U_1) = 1 - P(W_1) = 1 - 0.2 = 0.8\), where \(U_1\) is the event that a proposal is judged unworthy of funding by the first panel.

Calculate the probabilities of decision reversal given the proposals were worthy or unworthy of funding in panel 1

Since the second panel of experts reverses the decision to fund in \(30\%\) of the cases, we have: \(P(R|W_1) = 0.3\) and \(P(R|U_1) = 0.3\).

Calculate the probabilities of proposals being worthy or unworthy of funding in panel 2 given the decision in panel 1

Apply the conditional probabilities to obtain: \(P(W_2|W_1) = 1 - P(R|W_1) = 1 - 0.3 = 0.7\) (since the second panel reverses \(30\%\) of the cases), \(P(U_2|W_1) = P(R|W_1) = 0.3\), \(P(W_2|U_1) = P(R|U_1) = 0.3\), \(P(U_2|U_1) = 1 - P(R|U_1) = 1 - 0.3 = 0.7\).

Calculate the probabilities for the given scenarios

a. A worthy proposal is approved by both panels. We want to find the probability \(P(W_1 \cap W_2)\), using the formula \(P(W_1 \cap W_2) = P(W_2|W_1)P(W_1)\): \(P(W_1 \cap W_2) = P(W_2|W_1)P(W_1) = 0.7 \cdot 0.2 = 0.14\). b. A worthy proposal is disapproved by both panels. We want to find the probability \(P(W_1 \cap U_2)\), using the formula \(P(W_1 \cap U_2) = P(U_2|W_1)P(W_1)\): \(P(W_1 \cap U_2) = P(U_2|W_1)P(W_1) = 0.3 \cdot 0.2 = 0.06\). c. A worthy proposal is approved by one panel. We want to find the probability \(P(W_1 \cap W_2) + P(W_1 \cap U_2)\): \(P(W_1 \cap W_2) + P(W_1 \cap U_2) = 0.14 + 0.06 = 0.2\). So, the probabilities are: a. \(\boxed{0.14}\) for a worthy proposal being approved by both panels, b. \(\boxed{0.06}\) for a worthy proposal being disapproved by both panels, c. \(\boxed{0.2}\) for a worthy proposal being approved by one panel.

Key concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that deals with the likelihood of an event occurring, given that another event has already happened. For instance, if we want to know the chance that a research proposal gets approved by the second panel, knowing that it was previously deemed worthy by the first panel, we are actually inquiring about a conditional probability.
To calculate this, we use the formula:
\[\begin{equation} P(A|B) = \frac{P(A \cap B)}{P(B)}\end{equation}\]
where:

• P(A|B) is the conditional probability of event A occurring given that B has occurred.
• P(A \cap B) is the joint probability of both events A and B occurring.
• P(B) is the probability of event B.

In the exercise, the conditional probability for a proposal being approved by the second panel, given its approval by the first, was calculated. The explicit nature of how decisions from the first panel influence the second panel showcases the relevance of conditional probability in scenarios where outcomes are interdependent.

Example Application

Using the concept of conditional probability, we found that the probability a proposal that is worthy in panel one gets approved in panel two is 0.7, since 30% of decisions are reversed, thus:
\[\begin{equation} P(W_2|W_1) = 1 - P(R|W_1) = 1 - 0.3\end{equation}\]
This is a great example of how understanding conditional probability can lead to more informed decisions in real-world contexts such as funding allocation.

Complement of an Event

The complement of an event in probability is simply the probability that the event does not occur. It's a crucial concept since it offers a different perspective on likelihood and can simplify complex problems. If an event A has a probability P(A), then the complement of A, denoted as A', has a probability P(A') which equals 1 - P(A).
When dealing with complements, always remember that:

• The sum of the probabilities of an event and its complement is always 1.
• The complement helps calculate indirect probabilities by direct subtraction from 1.

In our exercise, we leveraged this concept to determine the chance that a proposal is deemed unworthy by the first panel, which is the complement of it being worthy:
\[\begin{equation} P(U_1) = 1 - P(W_1)\end{equation}\]
Understanding the relationship between an event and its complement is vital in probability and by extension in statistical reasoning.

Independent Events

In probability, independent events are those whose occurrence or non-occurrence does not affect the likelihood of another event happening. This concept is instrumental when the outcome of one event has no bearing on the outcome of another, and knowing one event occurred gives no information about whether the other will happen.
Two events A and B are independent if:
\[\begin{equation} P(A \cap B) = P(A) \cdot P(B)\end{equation}\]
For independent events, the probability of both events occurring simultaneously is the product of their individual probabilities. It's important to distinguish between independent events and mutually exclusive events. In the latter, if one event occurs, the other cannot, which differs from independence.
In our textbook problem, the key insight was assuming the panels function independently. If they didn't, we couldn't simply multiply the probabilities from each panel to get the joint probability. Instead, we’d need to use conditional probabilities. This difference is subtle but critical in correctly applying probability concepts to real-world scenarios.