Let's break this down step by step using the definitions and the given information:

Definitions and Given Information:

• Sensitivity: The probability of testing positive if the person is actually infected. Here, it's given as 95% (0.95).
• Specificity: The probability of testing negative if the person is not infected. Here, it's given as 98% (0.98).
• Prevalence: The proportion of the population that is actually infected. Here, it's given as 0.5% (0.005).

We are working with conditional probability and applying Bayes' Theorem for the positive predictive value.

1. Conditional Probability Analysis

a) What is the probability that a person tests positive given that they are actually infected?

This is directly the sensitivity of the test, as it gives the probability that an infected person tests positive.

$P(\text{Positive test} | \text{Infected}) = \text{Sensitivity} = 0.95$

So, the probability that a person tests positive given that they are infected is 95%.

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b) What is the probability that a person who tests negative is actually healthy?

This refers to the negative predictive value, i.e., the probability that a person who tests negative is not infected. This can be calculated using the formula:

$P(\text{Healthy} | \text{Negative test}) = \frac{P(\text{Negative test} | \text{Healthy}) \cdot P(\text{Healthy})}{P(\text{Negative test})}$

First, let's calculate the needed components:

• Specificity is the probability of a negative test when healthy, so: $P(\text{Negative test} | \text{Healthy}) = 0.98$

• Prevalence (the proportion of infected people in the population) is: $P(\text{Healthy}) = 1 - \text{Prevalence} = 1 - 0.005 = 0.995$

Now, we need to calculate the total probability of testing negative ($P(\text{Negative test})$):

$P(\text{Negative test}) = P(\text{Negative test} | \text{Healthy}) \cdot P(\text{Healthy}) + P(\text{Negative test} | \text{Infected}) \cdot P(\text{Infected})$

Where:

• $P(\text{Negative test} | \text{Infected}) = 1 - \text{Sensitivity} = 1 - 0.95 = 0.05$
• $P(\text{Infected}) = \text{Prevalence} = 0.005$

Now we compute $P(\text{Negative test})$:

$P(\text{Negative test}) = (0.98 \cdot 0.995) + (0.05 \cdot 0.005) = 0.9751 + 0.00025 = 0.97535$

Finally, we calculate the probability that a person who tests negative is actually healthy:

$P(\text{Healthy} | \text{Negative test}) = \frac{(0.98 \cdot 0.995)}{0.97535} = \frac{0.9751}{0.97535} \approx 0.9997$

So, the probability that a person who tests negative is actually healthy is approximately 99.97%.

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c) What is the probability that a person who tests positive is actually infected?

This is the positive predictive value, and we can calculate it using Bayes' Theorem:

$P(\text{Infected} | \text{Positive test}) = \frac{P(\text{Positive test} | \text{Infected}) \cdot P(\text{Infected})}{P(\text{Positive test})}$

Where:

• $P(\text{Positive test} | \text{Infected}) = \text{Sensitivity} = 0.95$
• (P(\text{Infected}) = \text