Answer to Question #176377 in Statistics and Probability for Zain

Question

2: 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. Report the probability that he will truly default. (6) 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 1

Hypothesis:

"H_1\\text{ --- "truly default"}\\\\\nH_2\\text{ --- "truly does not default"}"

Event "A\\text{ --- "student predicts positive"}"

$P(H_1)=\alpha\\ P(H_2)=100-\alpha\\ P(A|H_1)=0\\ P(A|H_2)=4\\ a)\ P(H_1|A)=\frac{P(H_1\cdot A)}{P(A)}=\frac{P(H_1)\cdot P(A|H_1)}{P(A)}=\\ =\frac{\alpha\cdot \alpha}{(100-\alpha)\cdot 4}=\frac{1}{4}\cdot \frac{\alpha^2}{100-\alpha}.\\ P(A)=P(H_1)P(A|H_1)+P(H_2)P(A|H_2)=\alpha\cdot 0+(100-\alpha)\cdot 4=\\ =(100-\alpha)\cdot 4.$

$b)$

$MX=-5000(100-\alpha)+20000\alpha=25000\alpha-500000.\\ MX=0\\ 25000\alpha-500000=0\\ \alpha=20\\ \alpha>20\%\text{ --- this is a good bet}.\\ \alpha\leq 20\%\text{ --- this is not a good bet}.$