Question

Suppose there is a gene which, if someone has it, causes them to turn into a werewolf if they are exposed to a full moon. Let W be someone having the werewolf gene. You know that P(W) = 0.3. One can test for the gene by taking someone's blood and dropping it on a sheet of silver and seeing if it reacts. Let S be this `silver test', and suppose that you are told that P(S) = 0.65. Suppose, finally, that you know that the `silver test' has a type-2 error rate of 0.05.

What is the type-1 error rate of the `silver test'?

Answer 1

Answer - We know that the `silver test' has a type-2 error rate of 0.05, which means that if someone has the werewolf gene, there is a 0.05 chance that the silver test will not detect it.

Let's use Bayes' rule to find the type-1 error rate of the `silver test'. We want to find P(S|~W), the probability that the silver test is positive given that the person does not have the werewolf gene.

P(S|~W) is the probability of a false positive, which is the type-1 error rate.

We know that P(W) = 0.3, which means that P(~W) = 0.7.

Let's assume that the `silver test' is independent of whether or not someone has the werewolf gene.

Using Bayes' rule, we have:

P(S|~W) = P(~W|S) * P(S) / P(~W)

We can find P(~W|S) using Bayes' rule:

P(~W|S) = P(S|~W) * P(~W) / P(S)

We know that P(S) = 0.65, P(W) = 0.3, and P(~W) = 0.7.

We can find P(S|W) using the type-2 error rate:

P(S|W) = 1 - type-2 error rate = 1 - 0.05 = 0.95

Using Bayes' rule, we have:

P(W|S) = P(S|W) * P(W) / P(S)

We can find P(S|~W) and P(~W|S) using the above equations:

P(S|~W) = P(~W|S) * P(S) / P(~W)
P(~W|S) = P(S|~W) * P(~W) / P(S)

Plugging in the values, we get:

P(S|~W) = P(~W|S) * P(S) / P(~W) = (1 - P(W|S)) * 0.65 / 0.7 = 0.54

P(~W|S) = P(S|~W) * P(~W) / P(S) = 0.54 * 0.7 / 0.65 = 0.58