Question 1185553
Here's how to break down this probability problem:

**a) Probability of True Default Given a Positive Prediction:**

We'll use Bayes' Theorem to solve this. 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 = 14% or 0.14.
* **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 (which we already know is just P(D)), or the student *doesn't* default *but* the test predicts positive (a false positive).

   * Probability of *not* defaulting = 1 - P(D) = 1 - 0.14 = 0.86
   * Probability of a false positive = 4% = 0.04
   * So, P(P) = P(D) + [P(not D) * P(false positive)] = 0.14 + (0.86 * 0.04) = 0.14 + 0.0344 = 0.1744

Now we can plug everything into Bayes' Theorem:

P(D|P) = (1 * 0.14) / 0.1744 ≈ 0.8026

Therefore, if the test predicts positive, there's approximately an 80.26% chance that the student will actually default.

**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) ≈ 0.8026.
* **Outcome 2: Student doesn't default:** You lose Rs. 5,000. The probability of this is 1 - P(D|P) ≈ 1 - 0.8026 = 0.1974.

Expected Value = (Probability of winning * Amount won) - (Probability of losing * Amount lost)
Expected Value = (0.8026 * 20000) - (0.1974 * 5000)
Expected Value = 16052 - 987
Expected Value = 15065

Since the expected value is positive (Rs. 15,065), this is a good bet for you to take *in the long run*.  Over many such bets, you would expect to profit.