Probability, Steroid Testing, &
Baseball
A CONDITIONAL PROBABILITY
EXERCISE
Question: What are the chances that a baseball player who does
not use steroids could have a "false positive" test
result?
A CONDITIONAL PROBABILITY EXERCISE
|
Question: What are the chances that a baseball player who does not use steroids could have a "false positive" test result? |
During the 2003 season of Major League Baseball (MLB), 7% of the major league baseball players tested positive for steroids. Steroids are a "performance-enhancing drug" and are banned in most professional and amateur sports. They are linked to causing serious long-term health problems for those who use them without a legitimate medical reason. Many fans of the game of baseball look at some of the recent home run records by "superstars" with a great deal of suspicion, wondering if the greatness displayed by the players are the results of talent and hard work or because they are "on the juice." In spite of this, the players and their union resist a strong testing program that would identify and punish those who cheat and would remove suspicion from the players who are clean.
Mathematics can play a role in the debate over testing for steroids. Testing for substances such as steroids involves taking a sample of the person's body (often blood, urine, or saliva) and analyzing if for the presence of the substance in the sample. If the substance is present in the sample, the person is said to test positive. If the substance is not present (or is present but at only a normal level), the person is said to test negative. However, the testing process is always subject to some degree of error. With enough practice, scientists can estimate the amount of errors that occur and thus determine a "percent of error." A test is valid only if the percent of error is low, but it is unlikely we will ever devise any test that is perfect. All tests will have some percent of error.
The current test for steroids claims to be accurate to less than one percent of error. Let us assume that the test is incorrect exactly 1% of the time it is administered and that there are in fact 7% of all baseball players using steroids. Let us also estimate that there are 1500 active players in a typical baseball season (about 30 teams with about 50 players over the course of a season).
[ Back to TOP ]
Question: What is the probability that, if a player tests positive for steroids, that the player is actually "clean?"
This is a conditional probability question. We need to determine the number of players who will test positive, how many of those are "clean," how many are on steroids, and so on. It is not a simple question. The method you could use to determine this is as follows.
1. How many players use steroids and how many are "clean?" For simplicity, let's call those who use steroids group S and those who do not, group C. The numbers of players in each group are straightforward percentages.
| Steroid-Users (S): | 7% of 1500:  |
| Players who are "clean" / non-users of steroids (C): | 93% of 1500:  |
2. Now we need to determine how many in each group (S and C) will test positive or negative for steroids. Note that it is possible to be "clean" and to falsely test positively for steroids. Similarly, steroid-users might test positive (accurately) or negative (falsely). Since the test is 99% accurate (or inaccurate 1% of the time), we will assume this accuracy applies to finding steroids when they are not there as well as not finding steroids when they are present.
a. How many steroid-users will test positive and how many will test negative falsely?
| Method | Total | |
| S who will test positive accurately |  |
103.95 |
| S who will test negative falsely |  |
b. How many clean players will test negative and how many will falsely test positive?
| Method | Total | |
|
C who will test negative accurately |
 |
|
|
C who will test positive falsely |
 |
3. If a player tests negative, since the test is 99% accurate, it will be assumed the player is "clean" and that player will no longer be tested. How many players (from either S or C) test negative and are judged as clean in this "first round" of testing?
( Clean and Neg ) + ( Steroid-user and Neg ) = Total who tested Negative
______________ + ___________________ = ___________________
4. If a player tests positive he will be labeled a "steroid-user." How many players (from either S or C) test positive and are judged as steroid-users in this "first round" of testing?
( Clean and Pos ) + ( Steroid-user and Pos ) = Total who tested Positive
______________ + ___________________ = ___________________
Question: What is the percent of players who test positive but are actually clean (meaning their test was a "false positive")?
Follow-up Question: If you were a MLB player, how would
you feel about mandatory testing for steroids?
[ Back to TOP ]
Second Round of Testing
"Get a second opinion." You must have heard that a thousand times when people talk about getting bad news from a doctor. The reason often has its roots in mathematics. What if all the players who tested positive for steroids said, "Not me, I'm clean. The test was faulty." The fact is that the test does produce false positive results. So, how do we trust any test? Answer: we take the test again and hope the test is correct the second time. Could the test be wrong twice in a row? Yes! This leads to our next question:
Question: What are the chances that a baseball player who does not use steroids could be tested twice and receive "false positive" test results both times?
To determine this, let us assume that every player who tested positive in the first round is required to take a second test for steroids.
5. How many from the steroid-user group, who tested positive once, will test accurately a second time and how many will test "false negative" in this second round? (We will assume any negative test result is accurate and thus the player will be freed from suspicion of steroid use.)
| Method | Total | |
| S who will test positive accurately |  |
|
| S who will test negative falsely |  |
6. How many from the clean group, who tested false positive in the first round, will test accurately a second time and how many will test "false positive" again in this second round?
| Method | Total | |
|
C who will test negative accurately |
|
|
|
C who will test positive falsely |
|
7. How many players (from either S or C) test negative and are judged as clean in the second round of testing?
( Clean and Neg ) + ( Steroid-user and Neg ) = Total who tested Negative
______________ + ___________________ = ___________________
8. How many players (from either S or C) test positive
and are judged as steroid-users because they tested positive
twice?
( Clean and Pos ) + ( Steroid-user and Pos ) = Total who tested Positive
______________ + ___________________ = ___________________
9. Question: What percent of all
positive results after the second round are actually "clean"
/ non-users of 
steroids?
[Express in both percent form and as a ratio. Further, express
your ratio "verbally;" for example, 60% is the same
as 60/100 = 3/5, verbally "3 out of every 5." Similarly,
1.5% = 1.5/100 = 15/1000, "fifteen out of every one-thousand."]
10. Question: What ratio of all baseball players in the total population will be "clean" / non-users of steroids but will test positive (falsely) twice? [Again, express your ratio numerically and "verbally."]
11. Question: What ratio of steroid-users in MLB will escape detection by mandatory testing of all players, with all who test positive to be tested a second time? [Express your ratio numerically and verbally.]
12. Question: If you were a MLB player, how would you feel about mandatory testing for steroids if any player who tests positive for steroids would be required to take a second test before any judgments were determined?
[ Back to TOP ]