Dear Math Students, My law firm represents a number of Olympic athletes that hav

Question

Dear Math Students,

My law firm represents a number of Olympic athletes that have been disallowed from competing in the Olympic games dues to failed drug tests. My clients assure me that they have never used performance-enhancing drugs, and yet a single failed drug test along with the Olympic Committee's zero-tolerance policy means that their careers have been significantly curtailed, if not ruined. My goal is to shed some sort of doubt on the results of the tests, followed by either an appeal to the committee or a civil suit if necessary. Unfortunately, my probability knowledge is not very good and so your help would be most appreciated.

The two key pieces of information are 1) the probability that the test is in error and 2) the probability that a given athlete is actually "doping." The committee sees a positive test and concludes that the athlete must have been doping, but as you well know nothing in life is perfectly certain and I would like to know the actual probability. The committee's stance is certainly strengthened by the fact that they perform both a urine test and a blood test. Studies have shown that the urine test is 90% accurate while the blood test is 98% accurate. However, the probability of 2) is much harder to pin down. Athletes fear that everyone is doping, but in reality, it is probably no more than 5%, especially since the committee claims they catch all dopers.

Given all this, what I need to know are the following:

1.) What is the probability a client was doping, given that they failed both tests?

2.) What is the probability a client was doping, given that they failed the urine test but passed the blood test?

Your help could make a large difference in many people's lives. I look forwarded to hearing from you, both about the mathematics and your advice on how to proceed.

Sincerely,

Larson Lawyerson

This is all the information I was given. It would be helpful if there was a brief explanation on how you got the answer.

Explanation / Answer

Suppose there are 1000 athletes. Consider the following table:

1)  The probability a client was doping, given that they failed both tests is:

(5/860)*(1/932)=0.000006

2) The probability a client was doping, given that they failed the urine test but passed the blood test is:

(5/860)*(49/68)=0.004189

Urine Test Doping Positive Negative Total Yes 50*.90=45 50-45=5 1000*.05=50 No 950-855=95 950*.90=855 950 Total 140 860 1000 Blood Test Doping Positive Negative Total Yes 50*.98=49 50-49=1 1000*.05=50 No 950-931=19 950*.98=931 950 Total 68 932 1000

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.