Calculating Conditional Probability for Steroid Use at the Olympics

[lina29]

Homework Statement

All athletes at the Olympic games are tested for performance-enhancing steroid drug use. The imperfect test gives positive results (indicating drug use) for 90% of all steroid-users but also (and incorrectly) for 2% of those who do not use steroids. Suppose that 5% of all registered athletes use steroids. If an athlete is tested negative, what is the probability that he/she uses steroids?

Homework Equations

The Attempt at a Solution

I did P(defective)=.05
P(Tested defective|defective)=1
P(Tested defective|good)=.02

From there I did P(TD)=(1*.05)+(.02*(1-.05))=.069

(1*.05)/.069=.724 which was wrong. Am I missing a step?

 

[Curious3141]

Homework Statement

All athletes at the Olympic games are tested for performance-enhancing steroid drug use. The imperfect test gives positive results (indicating drug use) for 90% of all steroid-users but also (and incorrectly) for 2% of those who do not use steroids. Suppose that 5% of all registered athletes use steroids. If an athlete is tested negative, what is the probability that he/she uses steroids?

Homework Equations

The Attempt at a Solution

I did P(defective)=.05
P(Tested defective|defective)=1
P(Tested defective|good)=.02

From there I did P(TD)=(1*.05)+(.02*(1-.05))=.069

(1*.05)/.069=.724 which was wrong. Am I missing a step?

First of all, sanity check. Doesn't that probability seem a tad high to you? We're talking about the probability that a negative test has mistakenly missed a steroid user. You're saying that almost 3/4 of those testing negative were actually using steroids. Even if the test is imperfect, it's not *that* imperfect. So you should realize your answer is way off.

I would suggest not renaming the categories to "defective" and "good" - it's confusing. I assume that when you wrote "defective", you meant "steroid user". So why is "P(Tested defective|defective)=1"? Shouldn't that be P(tested steroid positive|steroid user) = 0.9?

Also why are you considering the probability of having "tested defective" or tested positive for steroid use? You're asked about the scenario where the athlete tested negative. So all you need to consider are the scenarios that could have given you a negative test (or "tested good" using your label).

I've always found it helpful to draw a probability tree. You might want to do this. Start by having the first branch go to S+ and S- (for steroid user and steroid-nonuser, respectively). Each of those branches to T+ and T- (for steroid test positive and steroid test negative respectively). Now can you write down the probabilities at each branch?

EDIT: Attached an image, can you fill in the probabilities that go into each "?"