Question 1185546
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.