a) The cell corresponding to a positive diagnosis when the person has antibodies present as well as the cell for a negative diagnosis when the person has no antibodies present are the cells representing a CORRECT diagnosis.
The cell corresponding to a negative diagnosis when you actually have antibodies present is called a FALSE NEGATIVE and is considered as a TYPE II error.
The cell corresponding to a positive diagnosis when you actually don't have antibodies present is called a FALSE POSITIVE and is considered as a TYPE I error.
For the tree diagram for the questions below, refer to the picture attached.
b) To know the probability that a person without HIV will be diagnosed positive (false positive), we just trace the tree diagram from population to "antibodies not present" to "positive". The tree diagram will give us a value of 0.00588.
ANSWER: The probability of a false positive is 0.00588 or 0.588%.
c) We do the same thing as the previous subproblem to determine the probability that a person with HIV will be diagnosed as negative. We trace the tree diagram from population to "antibodies present" to "negative". The tree diagram will give us a value of 0.003.
ANSWER: The probability of a false negative is 0.003 or 0.3%.
d) Same thing for this subproblem. We trace the tree diagram from population to "antibodies present" to "positive" to know that the value is 0.01997.
ANSWER: The probability that a person with HIV will be diagnosed as positive is 0.01997 or 1.997%.