In the situation from the previous example, we want to find the probability that the diagnosis is incorrect. We know that according to the American Diabetes Association 8.3% of people in the United States have diabetes. The developed test is 98% accurate for people who have diabetes, and 95% accurate for people who do not have diabetes. Let's start by naming the occurring events.
A& - Person has diabetes
B& - Correct diagnosis
Note that when event A occurs P(B)=98%=0.98, and when event A does not occur P(B)=95%=0.95. The probability of event B depends on the occurrence of event A, so the events are dependent. Let's now consider the probabilities on a probability tree diagram. Note that the event of a person not having diabetes is a complement of event A, so we will mark it as A.
Since on the tree diagram there are complements of events A and B, we can use the formula for the probability of a complement to calculate their probability.
P(A)& =1- 0.083= 0.917
P(B|A)& =1- 0.98= 0.02
P(B|A)& =1- 0.95= 0.05
Let's add the obtained information to our diagram.
To find the probability that the diagnosis was incorrect we need to follow the branches of our diagram leading to event B. We have two ways leading to event B. We have to multiply the values in each way, and then add the values from both.
