Dependence
Goal: Start to consider dependence in probability
Objective: Practice conditional probability calculations
Example: Subsetting
Consider the months of the year
\[M = \{\text{Jan, Feb, Mar, Apr, May, Jun, Jul, Aug, Sep, Oct, Nov, Dec} \}\] Let us say that a month is ``long’’ if it has 31 days. What is the probability that we have a long month given that we are in the Spring semester?
\[S = \{ \text{Jan, Feb, Mar, Apr, May} \}\]
Conditional Probability
We can condense this process into a formula for conditional probability:
The conditional probability of observing event \(A\) given event \(B\) has already taken place is
\[P(A|B) = \displaystyle\frac{P(A \cap B)}{P(B)}\]
Example: Contigency Tables
In this hypothetical example, suppose that we are following an epidemiologist who is testing patients at a hospital in for the novel strain of coronavirus.
- Build a contingency table with the following data
- 175 true positives
- 32 false negatives
- 18 false positives
- 2019 true negatives
- Compute the probability that a randomly selected patient is disease free given that the drug test is positive.
Prosecutor’s Fallacy
- Using the same counts as the previous example, compute the probability that for a randomly selected patient the test returns positive given that the patient is disease free.
Converses for conditional probability are almost never equal.
\[P(A|B) \neq P(B|A)\]
What do we do when the order of the events switch?
Next time: Bayes’ Rule
Looking Ahead
- due Fri., Jan. 27:
- WHW2
- JHW0
- CLO (survey)
- and the before-lecture quizzes
Exam 1 will be on Wed., Mar. 1
