Question $27$

Contingency Tables and Conditional Probability: Homework and Passing a Course

A teacher categorized students by whether they regularly completed their homework $($or not$)$ and whether they passed the course $($or not$).$ The data are in the table below.

$\backslash$table$[[,$Completed HW$,$Did not Complete HW$,$Total$],[$Passed the Course,$23,8,31],[$Did not Pass the Course,$3,13,16],[$Total$,26,21,47]]$

A student from the class is randomly selected. Determine the following probabilities. Enter your results as percents rounded to two decimal places.

What is the probability a student passed the course?

What is the probability a student passed the course or completed the HW$?$

What is the probability a student passed the course and completed the HW$?$

What is the probability a student completed their homework given that they passed the course?