# Bayes Theorem

## Main Question or Discussion Point

Guys I am stuck with this can any one help me??/

Suppose that 2% of all Olympic athletes use steroids or other performance-enhancing drugs. Because drug
use is a serious violation of Olympic rules, the International Olympic Committee has implemented random
drug tests. Let us assume that the test will report positive with probability 0.95 if the athlete uses drugs,
and with probability 0.2 if the athlete does not use drugs.
a) A randomly selected athlete has a positive drug test. What is the probability that he uses drugs?
b) A randomly selected athlete has a negative drug test. What is the probability that she uses drugs?
c) In fact, an athlete is typically asked to give two samples, A and B, which are tested independently.
Assuming that the results of the A and B tests truly are independent, and that both of an athlete’s samples
are positive, what is the probability that he uses drugs?
d) Given that an athlete’s A test is positive, what is the probability that her B test will also be posi-
tive?

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Simon Bridge
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Welcome to PF;
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ok I am seriously not able to do much here but gonna give what all I could do

P(S)= prob of using steroids = 0.2
P(T|S) = Prob of testing positive given the athlete takes steroids
P(T|~S) = Prob of testing positive given the athlete doen not takes steroids

1) we need to find (T and S) = p(T|S)/p(S) = 0.019

2) we need to find (~T and S) = p(~T|S)/p(S) =??

3) ??
4) ??

Welcome to PF;
To get the most out of these forums, please show us your best previous attempt. This helps us target our response to your individual need.

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chiro
ok I am seriously not able to do much here but gonna give what all I could do

P(S)= prob of using steroids = 0.2
P(T|S) = Prob of testing positive given the athlete takes steroids
P(T|~S) = Prob of testing positive given the athlete doen not takes steroids

1) we need to find (T and S) = p(T|S)/p(S) = 0.019

2) we need to find (~T and S) = p(~T|S)/p(S) =??

3) ??
4) ??

Hey rohan.pitt and welcome to the forums.

Hint: What does P(T and S) + P(~T and S) equal to? (Think about what (T and S) OR (~T and S) in terms of sets and Venn diagrams)

Simon Bridge
Homework Helper
Sorry I could not respond right away - I have stuff to do like sleep, go to work, answer other peoples questions and so on ;) I'd had to wait 14 hours before you got back to me as it is .... looking at the time-stamp I was having breakfast with a beautiful woman about then - I'm sure you understand.

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Simon Bridge
Homework Helper
ok I am seriously not able to do much here but gonna give what all I could do

P(S)= prob of using steroids = 0.2
P(T|S) = Prob of testing positive given the athlete takes steroids
P(T|~S) = Prob of testing positive given the athlete doen not takes steroids
well P(S) is incorrect and you left off some values that you have been given:
according to the description in post #1
$P(S)=0.02$
$P(T|S)=0.95$
$P(T|\neg S) = 0.2$

The majority of this question is about how you turn the written description into the mathematical symbols ... so "the probability that a positive test means the athlete takes steroids" is written P(S|T) ... so how would you write "the probability that a negative test means the athlete is clean"?

After that it is a matter of just writing down Beyes' Theorem and putting the numbers in.

You can probably do all these without directly using Beyes' theorem by assuming you are testing some arbitrary number of athletes.
Say you have 1000 athletes to test:
1. how many do drugs?
2. how many of them will test positive?
3. how many don't do drugs?
4. how many of them test positive?
5. how many positive tests overall?
6. probability that a positive test means the athlete does drugs?
... rinse and repeat.

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