Question
1. Suppose you are the head of a laboratory that tests for a certain rare condition. Your test is pretty good: given that a patient actually has the condition, the patient wl (correctly) test positive 99% of the time! There are also very few false negatives: given that a patient does not have the condition, they will (correctly) test negative 98% of the time. However, the condition is very rare: only .1% of all people have this condition. So, given that someone tests positive, what is the probability that they actually have the condition? 2. Suppose you are taking an exam that adjusts its difficulty on later questions based on how well you do on previous questions. This exam has only two problems. The first problem is medium difficulty: you think you have only a 65% chance of answering it correctly. If you get the first problem correct. then it asks a more difficult question that you have a 40% chance of answering correctly. On the other hand, if you answer the first question incorrectly, then it will ask an easier question that you have a 85% chance of answering correctly What is the probability that you answer the second question correctly? 3. You know that 30% of students in a certain class have cats, while 70% of students in the same class have dogs. If the events of having a dog and having a cat are independent for students in this class, then what percentage of students have both a cat and a dog?
Explanation / Answer
1)probability of tested positive =P(have disease and tested positive+not have disease and tested posiitve)
=0.001*0.99+(1-0.001)*(1-0.98)=0.02097
therefore probability of having condition given tested postiive =0.001*0.99/0.02097 =0.04721
2)probability of answering second question correctly =answering first correct and then answering second difficult question correct+answering first incorrect and then answering second easy question correct
=0.65*0.40+(1-0.65)*0.85=0.5575
3)for independence; percentage of stiudents having cat and dog =(0.3*0.7)*100 =21%
