A doctor finds evidence of a serious illness in a particular patient and must make a determination about whether or not to advise the patient to undergo a dangerous operation. If the patient does suffer from the illness in question, there is a 95% probability that he will die if he does not undergo the operation. If he does undergo the operation, he has a 50% probability of survival. If the operation is conducted and it is discovered that the patient does not suffer from the illness, there is a 10% probability that the patient will die due to complications resulting from the operation. If it has been estimated that there is a 20% to 30% probability of the patient actually having the illness in question, how should the doctor advise her patient?
Several points to make here:
First, I think you may have slightly misread the information given. The doctor has detected *evidence* of the illness. That is not the same as saying the doctor has detected the illness where you might have to worry about false positives. The question is basically saying that based on the evidence the doctor has found, there is a 20-30% chance the patient actually has the illness.
Second, the reason the probabilities in the chance tree don't add up to 1 is because you have effectively given both "surgery" and "no surgery" a probability of 1 (think of your final probabilities as having a multiplied by 1 part corresponding to that part of the chart). If you were to split out the chart into one chart for "surgery" and one chart for "no surgery", you'd see ...
The probability evidence of a serious illness is determined.