Sally has caught covid but doesn’t know it

Answered question

2022-04-28

Sally has caught covid but doesn’t know it yet. She is testing herself with rapid antigen kits which have an 80% probability of returning a positive result for an infected person. For the purpose of this question you can assume that the results of repeated tests are independent. 

a) If sally uses 3 test kits what is the probability that at least one will return a positive result? 

b) In 3 tests, what is the expected number of positive results?

c) Sally has gotten her hands on more effective tests, these ones have a 90% probability of returning a positive result for an infected person. If she tested herself
twice with the new tests, how many positive results would she expect to see?

Answer & Explanation

alenahelenash

alenahelenash

Expert2023-05-02Added 556 answers

Let A be the event that Sally is infected with COVID-19 and B be the event that a rapid antigen test returns a positive result. We know that P(B|A) = 0.8 and that the results of repeated tests are independent.
a) We want to find the probability that at least one of the three tests will return a positive result, which is equivalent to finding the probability of the complement of the event that all three tests return negative results.
Using the complement rule, we have:
P(at least one positive result)=1P(all negative results)
Since the test results are independent, the probability of all three tests returning negative results is:
P(all negative results)=P(negative)3=(1P(positive))3=(10.8)3=0.008
Therefore, the probability that at least one of the three tests will return a positive result is:
P(at least one positive result)=10.008=0.992

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