Question 11

Scenario: If a resident of Santa Clara County tested positive for the SARS-CoV-2 antibody from that test, what is the probability that the person actually has the antibody?

Events:

  • - A randomly selected resident of Santa Clara County has the antibody to SARS-CoV-2.
  • - The BioMedomics Covid-19 IgM/IgG Rapid Test returns a positive result.

Step 1

Problem: Translate these probability values into probability notations:

  1. "2.8% of residents in Santa Clara County, California have antibodies to SARS-CoV-2."
  2. "The test correctly generates a positive result for people who have the condition that's being tested for 88.7% of the time."
  3. "The test correctly generates a negative result for people who don't have the condition that's being tested for 90.6% of the time."

Process: If 2.8% of residents have antibodies to SARS-CoV-2, then the probability that a randomly selected resident will have those antibodies is also 2.8%, or 0.028. We can rephrase the second sentence to be "The probability that the test generates a positive result for a resident given that they have the condition being tested for is 88.7%." This would be the conditional probability . We can rephrase the third sentence to be "The probability that the test generates a negative result for a resident given that they don't have the condition being tested for is 90.%."

Answer:

  1. .
  2. .
  3. .

Step 2

Problem: For each part of the tree diagram denoted in red numbers (1-6), write the correct probability notation and the correct probability value corresponding to it. For each part of the tree diagram denoted in blue roman numerals (I-IV), write the correct probability notation and the correct probability value corresponding to it.

A

A'

B

B'

B

B'

1

2

3

4

5

6

‎

I

II

III

IV

Process: ...

Answer:

  1. .
  2. .
  3. .
  4. .
  5. .
  6. .

I. .
II. .
III. .
IV. .

Step 3

Problem: Assign probability to each disjoint portion as denoted in the Venn diagram.

Process: ...

Answer:
7. 0.003164.
8. 0.024836.
9. 0.091368.
10. 0.880632.

Step 4

Problem: Write the desired probability in probability notation. Find the probability using the definition of the conditional probability and the fact that all four regions identified in Step 3 are disjoint (mutually exclusive).

Process: The question can be rephrased to ask what the probability is of a randomly selected person from Santa Clara Country having the antibody to SARS-CoV-2 given that they tested positive. This would be the conditional probability . We don't know the probability , but it can be found by adding the probabilities of the intersection of and and the intersection of the complement of and .

Answer: 0.213728.