Professor Frank Nabarro insists that all senior physics majors take his notorious physics aptitude test. The test is so tough that anyone not going on to a career in physics has no hope of passing, whereas 60% of the seniors who do go on to a career in physics still fail the test. Further, 76% of all senior physics majors in fact go on to a career in physics. Assuming that you fail the test, what is the probability that you will not go on to a career in physics Please, round your answer to two decimal places.
A)0.16
B) 0.34
C) 0.71
D) 0.66
E) 0.46

Question

Professor Frank Nabarro insists that all senior physics majors take his notorious physics aptitude test. The test is so tough that anyone not going on to a career in physics has no hope of passing, whereas 60% of the seniors who do go on to a career in physics still fail the test. Further, 76% of all senior physics majors in fact go on to a career in physics. Assuming that you fail the test, what is the probability that you will not go on to a career in physics  Please, round your answer to two decimal places.  
A)0.16
B) 0.34
C) 0.71
D) 0.66
E) 0.46

Quizplus · Accepted Answer

Let's denote:- $P(F)$ as the probability of failing the test.- $P(C)$ as the probability of going on to a career in physics.- $P(F|C)$ as the probability of failing the test given that a student goes on to a career in physics.- $P(F|\neg C)$ as the probability of failing the test given that a student does not go on to a career in physics.- $P(\neg C|F)$ as the probability of not going on to a career in physics given that a student fails the test.Given:- $P(C) = 0.76$, so $P(\neg C) = 0.24$ (since $P(C) + P(\neg C) = 1$).- $P(F|C) = 0.60$.- $P(F|\neg C) = 1$ (since anyone not going on to a career in physics has no hope of passing).We want to find $P(\neg C|F)$.First, we find $P(F)$ using the law of total probability:$$P(F) = P(F|C)P(C) + P(F|\neg C)P(\neg C) = 0.60 \times 0.76 + 1 \times 0.24 = 0.456 + 0.24 = 0.696$$Then, we use Bayes' theorem to find $P(\neg C|F)$:$$P(\neg C|F) = \frac{P(F|\neg C)P(\neg C)}{P(F)} = \frac{1 \times 0.24}{0.696} = \frac{0.24}{0.696}$$$$P(\neg C|F) = 0.34$$ (rounded to two decimal places).

Professor Frank Nabarro Insists That All Senior Physics Majors Take