A drug testing company advertises a test for steroids that is 93.5% effective at detecting steroid use. It has a true negative effectiveness of 93.5%. Suppose 8% of 10,000 professional cyclists are believed to be using steroids.
Suppose the haplessly named Lance Robustcalve tests positive for steroid use. What is the probability that he's really not using steroids?
To find the probability that Lance Robustcalve is really not using steroids given that he tested positive, we can use Bayes' theorem.
Let's define the events:
A: Lance Robustcalve tests positive for steroid use
B: Lance Robustcalve is not using steroids
We are given the following probabilities:
P(A|not B) = 93.5% (true negative effectiveness)
P(A) = 8% (prevalence of steroid use)
We need to find P(not B|A), the probability that Lance Robustcalve is not using steroids given that he tested positive.
Bayes' theorem states:
P(not B|A) = P(A|not B) * P(not B) / P(A)
To find P(not B), we can use the complement rule:
P(not B) = 1 - P(B)
The complement rule states that the probability of an event not happening (in this case, Lance Robustcalve using steroids) is equal to 1 minus the probability of the event happening.
Let's calculate step-by-step:
1. Calculate P(not B):
P(not B) = 1 - P(B) = 1 - 8% = 92% = 0.92
2. Plug in the values into Bayes' theorem:
P(not B|A) = P(A|not B) * P(not B) / P(A)
P(not B|A) = 93.5% * 0.92 / 8%
Remember to convert percentages to decimals by dividing by 100:
P(not B|A) = 0.935 * 0.92 / 0.08
P(not B|A) = 0.8607
So, the probability that Lance Robustcalve is really not using steroids given that he tested positive is approximately 0.8607 or 86.07%.
To calculate the probability that Lance Robustcalve is not using steroids given that he tested positive, we can use Bayes' theorem.
Let's define the events:
A: Lance Robustcalve is using steroids
B: Lance Robustcalve tests positive for steroid use
We want to find the conditional probability P(A' | B), which is the probability that Lance Robustcalve is not using steroids given that he tested positive.
According to Bayes' theorem:
P(A' | B) = (P(B | A') * P(A')) / P(B)
P(B | A') is the probability of testing positive given that Lance Robustcalve is not using steroids. In other words, it is the false positive rate, which is 100% minus the true negative effectiveness of the test, so it is 100% - 93.5% = 6.5%. Therefore, P(B | A') = 0.065.
P(A') is the probability that Lance Robustcalve is not using steroids, which is given as 1 minus the proportion of professional cyclists using steroids. So, P(A') = 1 - 0.08 = 0.92.
P(B) is the probability of testing positive, which can be calculated using the law of total probability:
P(B) = P(B | A) * P(A) + P(B | A') * P(A')
P(B) = (93.5% * 8%) + (6.5% * 92%) = 0.748 + 0.598 = 1.346
Now we can substitute the values into Bayes' theorem:
P(A' | B) = (0.065 * 0.92) / 1.346 ≈ 0.0449
Therefore, the probability that Lance Robustcalve is not using steroids given that he tested positive is approximately 0.0449 or 4.49%.
p = 100% - 8% = 92% = 0.92 = 92/100 = 23/25.
It doesn't matter how many others are using. You just want the probability that his test is false positive.
p = 100% - 93.5% = ?