Someone either has disease X or they are healthy. A test is used which is 90% sensitive and 80% specific (81.81% precise). The test is performed 8 times with different samples from the same person.
1) What is the rule to calculate the probability that someone has the disease given that 5 out of 8 tests came back positive?
2) More generally: that N out of M samples were positive.
3) Is there a way of calculating the confidence?