To solve this problem, we can use Bayes' Theorem, which provides a way to update the probability of an event based on new evidence.

Given:

• $P(I) = 0.30$ is the probability that a person is infected by the virus.
• $P(NI) = 0.70$ is the probability that a person is not infected by the virus.
• $P(\text{Positive Test} | I) = 0.80$ is the probability that the test is positive given the person is infected.
• $P(\text{Positive Test} | NI) = 0.15$ is the probability that the test is positive given the person is not infected.

We are asked to find the probability that a patient is not infected given a positive test result, i.e., we want to compute $P(NI | \text{Positive Test})$.

Bayes' Theorem:

Bayes' Theorem states:

$P(NI | \text{Positive Test}) = \frac{P(\text{Positive Test} | NI) \cdot P(NI)}{P(\text{Positive Test})}$

Where $P(\text{Positive Test})$ is the total probability of a positive test result, which can be found using the law of total probability:

$P(\text{Positive Test}) = P(\text{Positive Test} | I) \cdot P(I) + P(\text{Positive Test} | NI) \cdot P(NI)$

Step-by-step Calculation:

1. Calculate $P(\text{Positive Test})$: $P(\text{Positive Test}) = (0.80 \cdot 0.30) + (0.15 \cdot 0.70)$ $P(\text{Positive Test}) = 0.24 + 0.105 = 0.345$

2. Calculate $P(NI | \text{Positive Test})$: $P(NI | \text{Positive Test}) = \frac{(0.15 \cdot 0.70)}{0.345}$ $P(NI | \text{Positive Test}) = \frac{0.105}{0.345} \approx 0.3043$

Thus, the probability that the patient is not infected given that the test result is positive is approximately 0.3043 or 30.43%.

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Would you like a further breakdown or have additional questions?

Here are some related questions to expand on the topic:

1. How can Bayes' Theorem be applied to other types of diagnostic tests?
2. What happens if the probability of a false positive is reduced (i.e., $P(\text{Positive Test} | NI)$)?
3. How would the result change if the infection rate of the population increased to 50%?
4. What other factors might affect the reliability of diagnostic tests in real-world scenarios?
5. How can we assess the accuracy of a diagnostic test more comprehensively?

Tip: When using Bayes' Theorem in diagnostic problems, always ensure you correctly account for both the true positive and false positive rates.