Sure! Let's break this down step by step without using a tree diagram.

Key information:

• 15% of patients may have HIV, meaning there's an 85% chance a patient does not have HIV.
• If someone does not have HIV, the test is 98.5% accurate (meaning 98.5% of these tests give a correct negative result).

We are trying to find:

The probability that a patient tested negative and does not have HIV.

This can be thought of as the product of two things:

1. The probability that a patient does not have HIV (85%, or 0.85).
2. The probability that the test correctly identifies this patient as negative (98.5%, or 0.985).

So, the overall probability is:

$\text{Probability} = 0.85 \times 0.985$

Now, let's multiply that:

$0.85 \times 0.985 = 0.83725$

Answer:

The probability that a patient tested negative and is free of HIV is 0.83725, or about 83.7%.

Let me know if you'd like more details, or if you have questions about this! Here are 5 related questions that can help you explore this topic further:

1. What is the probability that a patient tests positive and actually has HIV?
2. What is the probability that a patient who tests negative actually has HIV (false negative)?
3. What is the probability that a patient who tests positive does not have HIV (false positive)?
4. How would the probability change if the clinic believed 20% of its patients had HIV instead of 15%?
5. What is the probability that a patient tests negative, whether or not they have HIV?

Tip: When calculating probabilities of multiple events happening together, multiplying the probabilities of each event helps you find the overall likelihood!