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You are 86% immune if you are fully vaccinated. RTPCR Test shows 95% accuracy. Find the probability that you are covid positive if you are fully vaccinated and your RTPCR test shows a negative report.

Manav Patel , 3 Years ago
Grade 12
anser 1 Answers
Askiitians Tutor Team

Last Activity: 12 Days ago

To determine the probability of being COVID positive given that you are fully vaccinated and your RTPCR test shows a negative result, we can use Bayes' theorem. This theorem helps us update our beliefs based on new evidence. Let's break down the problem step by step.

Understanding the Variables

First, let's define the terms involved:

  • P(V): Probability of being fully vaccinated = 1 (since we are considering only fully vaccinated individuals).
  • P(C): Probability of being COVID positive.
  • P(N | C): Probability of testing negative given that you are COVID positive (false negative rate).
  • P(N | V): Probability of testing negative given that you are fully vaccinated.

Given Information

From your question, we have:

  • Vaccination provides 86% immunity, meaning the probability of being COVID positive if vaccinated is 14% (P(C | V) = 0.14).
  • The RTPCR test has a 95% accuracy, which implies a 5% false negative rate (P(N | C) = 0.05).

Calculating the Probabilities

To find the probability of being COVID positive given a negative test result, we need to apply Bayes' theorem:

P(C | N) = (P(N | C) * P(C)) / P(N)

Step 1: Calculate P(N)

P(N) can be calculated using the law of total probability:

P(N) = P(N | C) * P(C) + P(N | ¬C) * P(¬C)

Where:

  • P(¬C) = 1 - P(C) = 0.86 (the probability of not being COVID positive).
  • P(N | ¬C) = 1 (assuming that if you are not infected, the test will always be negative).

Now substituting the values:

P(N) = (0.05 * 0.14) + (1 * 0.86)

P(N) = 0.007 + 0.86 = 0.867

Step 2: Calculate P(C | N)

Now we can substitute back into Bayes' theorem:

P(C | N) = (P(N | C) * P(C)) / P(N)

P(C | N) = (0.05 * 0.14) / 0.867

P(C | N) = 0.007 / 0.867 ≈ 0.00806

Final Interpretation

This means that the probability of being COVID positive, given that you are fully vaccinated and have received a negative RTPCR test result, is approximately 0.81%. This is a very low probability, indicating that a negative test result is quite reliable in this context, especially for fully vaccinated individuals.

In summary, while no test is perfect, understanding these probabilities can help you make informed decisions about your health and safety. If you have further questions or need clarification on any part of this process, feel free to ask!

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