Previously on Bayes’ Theorem…

• We used the law of total probability and Bayes’ Theorem to do some examples

  • Drawing cards from the top of standard deck.

  • COVID testing example

Take-home Message

Today, we will:

• More on the COVID testing example

• Define accuracy, precision, sensitivity, and specificity for a binary contingency table.

• How is Bayes’ Theorem is used?

Bayes’ Theorem

Image Source: “Bayes’ rule with a simple and practical example” by Tirthajyoti Sarkar

15.15-Minute Activity (1/2)

Suppose that two jars contain colored balls as described in the table below.

1. One jar is chosen at random and a single ball is selected. If the ball is white, what is the probability that it came from the 1st jar? Explain Why.
2. One jar is chosen at random and two balls are selected. If both balls are red, what is the probability it came from the 2nd jar? Explain why.
3. One jar is chosen at random and three balls are selected. If two balls are red and one is white, what is the probability it came from the 2nd jar? Explain why.

• Timer starts

15.15-Minute Activity (2/2)

• Solution for (a): \[P(J_1|W) = \frac{P(J_1)P(W|J_1)}{P(J_1)P(W|J_1) + P(J_2)P(W|J_2)} = \frac{ \frac{1}{2} \frac{3}{6} }{\frac{1}{2} \frac{3}{6} + \frac{1}{2} \frac{2}{6}} = \frac{3}{5}\]

• Solution for (b): \[P(J_2|R_2) = \frac{P(J_2)P(R_2|J_2)}{P(J_1)P(R_2|J_1) + P(J_2)P(R_2|J_2)} = \frac{\frac{1}{2} \frac{4}{6} \frac{3}{5}}{\frac{1}{2} \frac{3}{6} \frac{2}{5} + \frac{1}{2} \frac{4}{6} \frac{3}{5}} = \frac{2}{3}\]

• Solution for (c): \[P(J_2|R_2 W) = \frac{P(J_2)P(R_2W|J_2)}{P(J_1)P(R_2 W|J_1) + P(J_2)P(R_2 W|J_2)} = \frac{4}{7}\]

For a more detailed solution see this pdf notes.

COVID Testing Accuracy

Example: There is no test that is 100% accurate in detecting COVID. Suppose that there is a 95% percent accuracy in detecting COVID infections.

What does accuracy mean?

Useful terms:

• True positive: Person test positive with the actual COVID.
• False positive: Person tests positive without the actual COVID.
• False negative: Person tests negative with the actual COVID.
• True negative: Person tests negative without the actual COVID.

Accuracy vs Precision

100,000 Patients Example

Suppose that we have \(100,000\) patients with \(1000\) patients are infected and \(99,000\) patients are not, and below shows how many of them tested positive and tested negative. Given this example, the probability of having actual COVID is 1% (which is also known as the prevalence rate).

Sensitivity and Specificity

• Sensitivity: the probability of a person tests positive with the actual COVID (this is the true positive rate) \[P(+|C^+) = 0.95\]

• Specificity: the probability of a person tests negative without the actual COVID (this is the true negative rate) \[P(-|C^-) = 0.95\]

• How is this information useful? Suppose we randomly select a person from a population and they tested positive. What is the probability that they have COVID? Here, we want to compute the conditional probability \[P(C^+|+).\]

Summary

Today, we discussed the following:

• Defined accuracy, precision, sensitivity, and specificity on a binary contingency table

• How is Bayes’ Theorem used? The balls in jars (also balls in urns) are a simplified way to illustrate conditional probabilities and Bayes’ Theorem, which is applied to the COVID testing example.

Next, we will discuss:

• Continuing on the COVID testing example and Bayes’ Theorem