Question

A company is interviewing potential employees. Suppose that each candidate is either qualified, or unqualified with given probabilities q and 1 − q, respectively. The company tries to determine a candidate’s qualifications by asking 20 true-false questions. A qualified candidate has probability p of answering a question correctly, while an unqualified candidate has a probability p of answering incorrectly. The answers to different questions are assumed to be independent. If the company considers anyone with at least 15 correct answers qualified, and everyone else unqualified, give a formula for the probability that the 20 questions will correctly identify someone to be qualified or unqualified.

Answer 1

Answer:

$P(C=1|T=1)=q(\sum_{i=15}^{20}\binom{20}{i} p^i(1-p)^{20-i})( \sum_{i=15}^{20}\binom{20}{i}[qp^i(1-p)^{20-i} + (1-q)p^{20-i}(1-p)^i])^{-1}$

Step-by-step explanation:

Hi!

Lets define:

C = 1  if candidate is qualified

C = 0 if candidate is not qualified

A = 1 correct answer

A = 0 wrong answer

T = 1 test passed

T = 0 test failed

We know that:

$P(C=1)=q\\P(A=1 | C=1) = p\\P(A=0 | C=0) = p$

The test consist of 20 questions. The answers are indpendent, then the number of correct answers X has a binomial distribution (conditional on the candidate qualification):

$P(X=x | C=1)=f_1(x)=\binom{20}{x}p^x(1-p)^{20-x}\\P(X=x | C=0)=f_0(x)=\binom{20}{x}(1-p)^xp^{20-x}$

The probability of at least 15 (P(T=1))correct answers is:

$P(X\geq 15|C=1)=\sum_{i=15}^{20}f_1(i)\\P(X\geq 15|C=0)=\sum_{i=15}^{20}f_0(i)$

We need to calculate the conditional probabiliy P(C=1 |T=1). We use Bayes theorem:

$P(C=1|T=1)=\frac{P(T=1|C=1)P(C=1)}{P(T=1)}\\P(T=1) = qP(T=1|C=1) + (1-q)P(T=1|C=0)$

$P(T=1)=q\sum_{i=15}^{20}f_1(i) + (1-q)\sum_{i=15}^{20}f_0(i)\\P(T=1)=\sum_{i=15}^{20}\binom{20}{i}[qp^i(1-p)^{20-i} + (1-q)p^{20-i}(1-p)^i)]$

$P(C=1|T=1)=q(\sum_{i=15}^{20}\binom{20}{i} p^i(1-p)^{20-i})( \sum_{i=15}^{20}\binom{20}{i}[qp^i(1-p)^{20-i} + (1-q)p^{20-i}(1-p)^i])^{-1}$