Probability Problem

Discus: SAT/ACT Tests and Test Preparation: October 2003 Archive: Probability Problem
 By Phantom (Phantom) on Monday, September 29, 2003 - 04:17 pm: Edit

The following was an extra credit problem:

There was a group of 10,000 athletes who needed to be tested for steroid use. 500 of the athletes do actually use steroids. The steroid test is only 95% accurate. An athlete was chosen at random to take the test; the result said that he did not take steroids. What is the probability that the athlete does actually use steroids?

How do you do it and why?

The answer I wrote down was wrong (though I still think it's right), so I'd like to see what you think about it. I will post the answer here in an hour.

 By Geniusash (Geniusash) on Monday, September 29, 2003 - 04:36 pm: Edit

9500(.95)+500(.05)=group that would test negative for steroids.
500(.05)=group that tests negative but takes steroids so,

 By Phantom (Phantom) on Monday, September 29, 2003 - 04:41 pm: Edit

5/1802 is wrong, sorry. There's no calculus involved. I'll post the answer after I get a few more replies, though I still do not understand it.

 By Geniusash (Geniusash) on Monday, September 29, 2003 - 04:49 pm: Edit

No, I did not have a calculator Im just not so good at mental math, the answer *with a calculator* should be 1/362 or .27624309% if I'm not just being stupid

 By Phantom (Phantom) on Monday, September 29, 2003 - 04:49 pm: Edit

Actually, I'll tell you what my answer is, but it's wrong. I thought that it was a trick problem and that the answer would be 500/10000, or 1/20 (5%). I felt that the whole test/95% accuracy thing was excess information (but apparently not).

If you ignore the test part, it's just:

There was a group of 10,000 athletes who needed to be tested for steroid use. 500 of the athletes do actually use steroids. An athlete was chosen at random to take the test. What is the probability that the athlete uses steroids?

The answer would be 500/10000. So how would the test affect the probability of whether the athlete chosen at random takes steroids or not? Wouldn't there always be a 1 in 20 chance the athlete does take steroids?

 By Geniusash (Geniusash) on Monday, September 29, 2003 - 04:54 pm: Edit

Your population is actually just the people that would test NEGATIVE for steroids (since you know the athlete tested negative). That is 95% of the non-users (.95(9500)) and 5% of the user group (.05(500)). By adding those together you get the entire population (9,050 people). You are only looking for an athlethe who would test negative, but actually take steroids so thats 5% of the users (25). So, 25/9050 should be your answer. (1/362)

 By Phantom (Phantom) on Monday, September 29, 2003 - 04:58 pm: Edit

nope your answer is wrong, but I understand what you're saying and I understand why my thinking was faulted now, thanks.

How on earth do you get such a large number?!

 By Xiggi (Xiggi) on Monday, September 29, 2003 - 05:00 pm: Edit

That is weird problem to begin with.

You are given a Total population of 10,000 and a level of confidence of 95% but the size of the sample is not given.

Is this from a math class or a stat class?

 By Phantom (Phantom) on Monday, September 29, 2003 - 05:03 pm: Edit

it's a precalculus math class bonus problem

 By Geniusash (Geniusash) on Monday, September 29, 2003 - 05:05 pm: Edit

Was this an extra-credit problem on a test? Cuz I'm like 100% this is the answer, unless the stroid test is only 95% accurate for the user group, and 100% accurate for the non-users in which case the answer would be 25/9525 or 1/381, in which case this problem is poorly worded. If, for some reason, the answerer did not omit the people who failed the test, the answer would be 25/10000, but that is really stupid. What's the answer?

 By Geniusash (Geniusash) on Monday, September 29, 2003 - 05:08 pm: Edit

50% is not right.

 By Phantom (Phantom) on Monday, September 29, 2003 - 05:08 pm: Edit

The answer was 50% (I edited it in in my last post.) I have no idea how it was obtained--your method seemed logical to me. The teacher did say that the answer was very surprising, even to him. It was an extra credit homework problem.

 By Geniusash (Geniusash) on Monday, September 29, 2003 - 05:10 pm: Edit

I think you may have switched the question around! Should it be, that he didn't take steroids but yet tested positive, cuz that IS a 50% chance

 By Phantom (Phantom) on Monday, September 29, 2003 - 05:11 pm: Edit

If no one else has an opinion on this, I think I'll just let this post die down until I can ask my teacher tomorrow about it. I'll share the answer with you guys when I find out. I really don't get how it can be 50%!!!

 By Geniusash (Geniusash) on Monday, September 29, 2003 - 05:13 pm: Edit

It can't the question should read..."There was a group of 10,000 athletes who needed to be tested for steroid use. 500 of the athletes do actually use steroids. The steroid test is only 95% accurate. An athlete was chosen at random to take the test; the result said that he DID(did not) take steroids. What is the probability that the athlete does NOT actually use steroids?"

 By Phantom (Phantom) on Monday, September 29, 2003 - 05:15 pm: Edit

the question was: "what is the probability that the athlete does use steroids?" I am positive about that (that's why I said 1/20 in my original answer). I am also really sure that the test result came up negative (i don't have the paper with me right now) which is why I don't understand how the answer can be so large. I think we should hold off discussion until tomorrow so i can double check.

 By Chobo (Chobo) on Monday, September 29, 2003 - 05:22 pm: Edit

The correct answer is 50%. If you still need a solution I can post it for you.

 By Phantom (Phantom) on Monday, September 29, 2003 - 05:30 pm: Edit

yeah, I still need the solution...can you please post it? thanks very much

 By Xiggi (Xiggi) on Monday, September 29, 2003 - 05:40 pm: Edit

It is not surprising considering that the test is 95% accurate for 10,000 athletes and the sample is only 1. This exercise is probably an introduction to the concept of "margin or errors" and "confidence interval". At 95%, it is about twice the standard deviation.

 By Digmedia (Digmedia) on Monday, September 29, 2003 - 06:30 pm: Edit

The test is 95% accurate; a person being tested tests negative; thus there is a 95% chance that the person does not use steroids. That leave 5% chance that the test is wrong.

The 10,000 is not the sample for determining the drug's accuracy. It is only the group from which a person was chosen. You are GIVEN the accuracy of the test, and thus, that is the accuracy.

If the answer is anything other than 5% then the stated accuracy of the test is incorrect.

 By Xiggi (Xiggi) on Monday, September 29, 2003 - 10:08 pm: Edit

Digmedia~

It does not work like that. There is a relation between the margin of error and the size of the SAMPLE. The sample is NOT 10,000.

 By Y17k (Y17k) on Tuesday, September 30, 2003 - 07:46 am: Edit

conditional probability...

Pr(Athelete uses steroids | the test said he doesnt take steriods) = 0.05/0.95 = 1/19

or 5.263 %

 By Xiggi (Xiggi) on Tuesday, September 30, 2003 - 11:42 am: Edit

Again, it does not work like that.

Here's how the "50-50" answer is determined:

1. The population consists of 10,000 people. Of those, we assume for this problem that 95% (9500) are nonusers and that 5% (500) are users.

Of the 9500 nonusers, 95% (9025) will test negative. That means 5% (475) will test positive. Of the 500 users, 95% (475) will test positive. That means that 5% (25) will test negative.

These are the totals:

9025 true negatives (nonusers)
475 false positives (nonusers)
475 true positives (users)
25 false negatives (users)

There are 475 "false positives" and 475 "true positives" -- a total of 950 positives -- so when there's only a 50-50 chance she or he is a user.

 By Xiggi (Xiggi) on Tuesday, September 30, 2003 - 12:10 pm: Edit

Could it be that the teacher wrote the problem backwards?

 By Geniusash (Geniusash) on Tuesday, September 30, 2003 - 03:27 pm: Edit

That's what I think

 By Digmedia (Digmedia) on Tuesday, September 30, 2003 - 04:16 pm: Edit

Sorry, but I don't get it. The problem asks, in effect, what is the probability of a false negative? By your reasoning, that's only 25 out of the 9050 potential negatives (geniusash's answer above).

The reasoning that says, from a population of 95% of non-users, a person is picked at random and tests negative, that that leads to a probability of 50% that the person IS a user does not make sense under ANY circumstances.

If you take a person from that group and do NO testing, there is a 95% chance that he or she is NOT a user. Now you apply a test that further indicates (with 95% accuracy) that the person is NOT a user. I don't care what formula you use, 50% is not even in the same universe of possibilities.

 By Digmedia (Digmedia) on Tuesday, September 30, 2003 - 04:26 pm: Edit

Never mind, I see now rereading your solution that you are assuming the question was written incorrectly, and are solving the corrected-statement problem.

The most interesting thing about this problem are the social implications. If you get a random drug test at school and you test positive (under a similar statistical environment), can you argue that there is only a 50/50 chance that you really are a user? I think the administration would fall back onto the "95% accuracy" thing and still ream you a new you-know-what.

 By Xiggi (Xiggi) on Tuesday, September 30, 2003 - 06:30 pm: Edit

Digmedia~

Yes, I tried to illustrate the 50% answer.

Interestingly enough, there has been a debate about randon drug testing. Opponents of the tests have used the conclusion that there are as many FALSE positive as TRUE positive if the tests have a 95% accuracy. The proponents of the tests argue that taking the test again would almost eliminate the chance of a false positive. The opponents are claiming that retaking the same test would simply be two independent events and that the probability remains the same.

The sad part is that innocent people have been victimized by drug-tests -truck drivers, for instance- and have had a difficult time removing a false positive from their records.

The only way to reduce the problem is to have tests that have a higher accuracy than the expected test value of the population. In other words, if the expected ratio of non-users is 95%, the accuracy of the test has to exceed 95%. Otherwise, false positives will always equal true positives.

I still think it is a weird problem

 By Geniusash (Geniusash) on Wednesday, October 01, 2003 - 03:54 pm: Edit

What's up with this prob?

 By Digmedia (Digmedia) on Wednesday, October 01, 2003 - 06:03 pm: Edit

Dunno, but this problem absolutely facinated me, mostly because of the real-world implications re drug tests, etc.

I've passed the problem on (in its corrected form) to several people and the response has been, like, "that can't be..."

 By Phantom (Phantom) on Wednesday, October 01, 2003 - 06:13 pm: Edit

Actually, I did copy the problem correctly onto my first post...I'll check with my teacher again tomorrow, I think the problem itself was backwards.

Anyway, thanks everyone for all your interest in my problem, especially Xiggi who posted the solution to the problem and also Geniusash for his interest.

 By Geniusash (Geniusash) on Wednesday, October 01, 2003 - 07:59 pm: Edit

A-hem, *she*. Why is it I always come off so masculine?

 By Fairyofwind (Fairyofwind) on Wednesday, October 01, 2003 - 08:07 pm: Edit

Post a pic in your profile and you won't You'd think you were Pokemon collector Ash

 By Geniusash (Geniusash) on Wednesday, October 01, 2003 - 09:12 pm: Edit

Actually, it's Ash-ley.

 By Digmedia (Digmedia) on Thursday, October 02, 2003 - 10:45 am: Edit

I went through this problem with the principal of our HS yesterday and he was facinated by it also. He wants me to email him the problem and solution. Thanks Phantom for posting it originally.

When you see the problem originally, you think this is easy but then you think it's hard (it was for me). Then, when you see the solution, it's like that WAS easy; why was it so hard to see the solution the first time>

 By Phantom (Phantom) on Thursday, October 02, 2003 - 04:26 pm: Edit