Chapter 3 M3: Probability

In this module, we’re looking at probability – with the goal of gaining both formal mathematical tools and some intuition for how things work. First, we think about random processes and the probability of various events; then we proceed to more advanced rules for working with event probabilities, plus probabilities for quantitative random variables. Some key skills are:

• Define a random process, outcome, and event; give examples of each, and identify examples given to you
• Write out the addition rule (which comes in two versions, generalized and limited) and multiplication rule (also generalized and limited)
  • Determine which rule, if any, to use for a given problem or question
  • Describe any special assumptions/conditions that have to be true to use the rule, and determine if those assumptions are valid in a given example
• Interpret key probability visualizations (and draw one yourself given the right information!)
  • Venn diagrams
  • Contingency tables
• Define conditional, marginal, and joint probabilities; give examples of each, and identify examples given to you
  • For example, in the context of studying students’ majors and their handedness, I might ask “What kind of probability question is ‘what percentage of science majors are left-handed?’” (and I’d also expect you to be able to answer the probability question, as long as you had enough information!)
  • …or I might ask “What’s a joint probability question you could ask in this context?”
• Write out Bayes’ rule, apply it in a given problem, and interpret its results in context
• Explain what a probability distribution is, in general
• Interpret the probability distribution of a random variable (either categorical or quantitative!) given in table form, or write the distribution yourself if given sufficient information
• Write the equation for expected value and explain what the pieces represent
• Write the equation for (theoretical) variance and explain what the pieces represent
• Write the rules for what happens to expectation and variance when we combine random variables (scaling by a constant, adding a constant, adding RVs together)
  • …and match these rules to context! When should we use each rule? What do we plug in where?
• Explain what a density function is, more or less
  • This module doesn’t really give you enough info (or time) to do in-depth work with density functions; we’ll do more with that in the next module.
  • The important thing right now is to understand why we can’t just write a continuous variable’s distribution in table form :)

This is a good moment for a reminder that this is not an arithmetic class :) What I’m always looking for is your ability to set up calculations and match formulas to the problem context/interpretation. That’s your human job – computers can do the rest! So on an Assessment, it’s always okay to write down the calculation you would do to answer a question, without working out all the arithmetic.