SOLUTION: Defaulting on a loan means failing to pay it back on time. The default rate among X university students on their student loans is α %. As a project you develop a test to predict

Question 1185546: Defaulting on a loan means failing to pay it back on time. The default rate among X university
students on their student loans is α %. As a project you develop a test to predict which students
will default. Your test is good but not perfect. It gives 4% false positives, i.e. prediciting a
student will default who in fact will not. If has a 0% false negative rate, i.e. prediciting a student
won't default who in fact will.
a) Suppose a random student predicts positive. What is the probability that he will truly
default.
b) Someone offers to bet me the student in part(a) won't default. They want me to pay them
Rs. 5k if the student doesn't default and they'll pay me Rs. 20k if the student does default.
Is this a good bet for me to take?
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to analyze this situation:
**a) Probability of True Default Given a Positive Prediction:**
We'll use Bayes' Theorem. Let:
* D be the event that a student defaults.
* P be the event that the test predicts positive (a student will default).
We want to find P(D|P), the probability that a student defaults given a positive prediction. Bayes' Theorem states:
P(D|P) = [P(P|D) * P(D)] / P(P)
Let's break down the components:
* **P(D):** This is the default rate, given as α% or α/100.
* **P(P|D):** This is the probability of a positive prediction given that the student defaults. Since there are no false negatives, if a student defaults, the test will *always* predict positive. So, P(P|D) = 1.
* **P(P):** This is the probability of a positive prediction. This can happen in two ways: either the student defaults *and* the test predicts positive, or the student *doesn't* default *but* the test predicts positive (a false positive).
* Probability of *not* defaulting = 1 - P(D) = 1 - (α/100)
* Probability of a false positive = 4% = 0.04
* So, P(P) = P(D) + [P(not D) * P(false positive)] = (α/100) + [(1 - α/100) * 0.04]
Now we can plug everything into Bayes' Theorem:
P(D|P) = [1 * (α/100)] / [(α/100) + (0.04 - 0.04α/100)]
P(D|P) = α / (α + 4 - 0.04α)
P(D|P) = α / (0.96α + 4)
**b) Is the Bet a Good One?**
To determine if the bet is good, we need to calculate the expected value of the bet.
* **Outcome 1: Student defaults:** You win Rs. 20,000. The probability of this is P(D|P) = α / (0.96α + 4).
* **Outcome 2: Student doesn't default:** You lose Rs. 5,000. The probability of this is 1 - P(D|P) = 4 / (0.96α + 4).
Expected Value = (Probability of winning * Amount won) - (Probability of losing * Amount lost)
Expected Value = [α / (0.96α + 4) * 20000] - [4 / (0.96α + 4) * 5000]
Expected Value = [20000α - 20000] / (0.96α + 4)
**Is it a good bet?**
* If the Expected Value is *positive*, it's a good bet for you in the long run.
* If the Expected Value is *negative*, it's a bad bet for you.
You need to plug in the value of α to determine the numerical value of the Expected Value.
**Example (Let α = 14):**
P(D|P) = 14 / (0.96*14 + 4) = 14 / 17.44 ≈ 0.8026 (as calculated before)
Expected Value = (0.8026 * 20000) - (0.1974 * 5000) = 16052 - 987 = 15065
In this example, the expected value is positive, so it's a good bet. You'll need to do the calculation with your specific value of α to know for sure.

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