This is a question I came across recently, any help would be greatly appreciated!:
A biostatistician is testing genes to determine their association to a certain disease. Each test for a given gene has a $5$% chance of a false positive for association (that is, about $5$% of tests performed on genes with no association will have an outcome showing association).
The biostatistician performs the test on $4$ separate genes so that the outcome of each test is independent (this means that the outcome of different tests don’t affect one another).
$(a)$ Define a suitable outcome space for this experiment.
$(b)$ Define an appropriate probability function for the outcome space of Part $(a)$ to answer the following question: If in reality, none of the 4 tested genes have an association with the disease, what is the chance that at least one of the tests will (falsely) show an association?
I am not sure about several things:
$1.$ Would the outcome space be $(+,-)$ for the test (testing pos. and neg.) or would it be all the combinations for the $4$ subjects testing positive and negative? (I am leaning towards the second one but would just like to make sure)
$2$. I understand that we have been given that $P(+|D') = 0.05$ and it is my understanding that we are being asked to find $P(X \geq 1)$ where $X$ is the number of false positives but I am not sure how to go about this at all.
Assigning $P(D) = 0.01$ and simply calculating $4 * 0.01 * 0.05$ seems too simple and wrong.
I'm really struggling with probability concepts so I would be grateful for any explanations!