Conditional Probability and Bayes’ Theorem

Videos (~40 mins)

Conditional Probability With Tables | Chance of an Orange M&M??? (9:37)
Bayes’ Theorem - The Simplest Case (5:31)
Bayes’ Theorem Example: Surprising False Positives (12:36)
Bayes’ Theorem - Example: A disjoint union (8:32)

Optional

Bayes theorem, the geometry of changing beliefs by 3blue1brown (15:10)

Reading (~30 mins)

Discrete Mathematics and Its Applications Section 7.3, pages 468-475

Warmup (~40 mins)

Problem 1

Your friend has two coins. One of them (coin \(A\)) has probability of heads equal to \(\frac{1}{2}\), while the other one (coin \(B\)) has probability of heads equal to \(\frac{3}{4}\). Your friend picks one of the two coins with probability \(\frac{1}{2}\), and then flips it 5 (independent) times. Find the probability that your friend chose coin \(A\), given the results, where the results are:

\(HHHHH\)
\(HTTTH\)
\(HHTTH\)

Problem 2

Preventing, experiencing, and managing COVID-19 is, regrettably, now a part of our everyday lives. One of the most important ways we can manage COVID-19 is by testing on a regular basis.

Please be aware that all numbers in this problem are made up unless I specifically cite them. This problem does not constitute medical or preventative care advice.

Part A

Suppose that 5% of the population has the common cold, 3% have COVID-19. You can interpret this as the probability that a randomly selected person has a 5% chance of having the common cold and a 3% chance of having COVID-19.

Assume that the short-term symptoms of these two illnesses are identical, that they have equal rates of transmission, that everyone who has either illness always has the sniffles, that it is impossible to have COVID and the common cold at the same time, and that there are no other illnesses that give you the sniffles.

You wake up one morning and you have the sniffles. What is the probability that you have COVID-19, given that you have the sniffles?

Part B

Many of us use rapid antigen tests to assess whether or not we have COVID-19. The false positive rate of a rapid antigen test is the probability that the antigen test returns a positive result given that you do NOT have COVID-19. So, if \(+\) is the event that the test returns a positive result and \(C\) is the event that you have COVID-19, the false positive rate is \(p(+|\bar{C})\). The false negative rate of a rapid antigen test is \(p(-|C)\), the event that the test incorrectly returns a negative result when you do in fact have COVID-19. The true positive rate is \(p(+|C)\) and the true negative rate is \(p(-|\bar{C})\).

Popular rapid antigen tests have a true positive rate for symptomatic individuals of approximately 73% and a true negative rate of approximately 99.6%.¹ The false positive rate is approximately 0.4% and the false negative rate is approximately 27%.

¹ https://www.cochrane.org/CD013705/INFECTN_how-accurate-are-rapid-antigen-tests-diagnosing-covid-19

Suppose that you wake up one morning with the sniffles. Because you are very responsible, you take a rapid antigen test.

The test comes up negative. What is the probability that you have COVID-19, given that you have a negative test and the sniffles? This is \(p(C|-\cap S)\). Hints:
- The way to approach this is to use Bayes’ rule to write \(p(C|-\cap S) = \frac{p(-\cap S | C)p(C)}{p(-\cap S|C)p(C) + p(-\cap S| \bar{C})p(\bar{C})}.\) Then, compute each of the terms appearing in this fraction. Because you always have the sniffles when you have COVID-19, \(p(-\cap S|C) = p(-|C)\).
- You also need to calculate \(p(-\cap S|\bar{C})p(\bar{C})\). You may assume that \(p(-\cap S|\bar{C}) = p(-|\bar{C})p(S|\bar{C})\). This is called conditional independence, an important topic which we sadly won’t hvae time to discuss.
The test comes up positive. What is the probability that you have COVID-19, given that you have a positive test and the sniffles?
- This problem is very similar to the previous one. You may assume that \(p(+\cap S|\bar{C}) = p(+|\bar{C})p(S|\bar{C})\).