Probability, uncertainty and expected outcomes
Count relevant outcomes, distinguish conditional and independent events, update beliefs from base rates and compare random outcomes by expectation and variation.
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Recommended preparation and ways to catch up
- A probability is between zero and one; mutually exclusive exhaustive outcomes sum to one.
- Before counting: decide whether order matters and whether items can be selected again.
- Before conditioning: name the reference set after the word “given”; it can change the denominator.
- Before expectation: a random variable maps outcomes to numbers, and a weighted average need not itself be a possible outcome.
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Your units and lessons
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Count and combine events
Enumerate ordered versus unordered outcomes and check the assumptions behind multiplying probabilities.
Count teams and ordered roles
When does order create a different outcome?
Check whether conditioning changes a probability
Are independence and mutual exclusion the same?
Condition and update
Use trees and Bayes’ rule to distinguish likelihood from posterior probability and handle impossible conditioning events.
Reverse a probability question
Does being late mean it probably rained?
Update a belief from counts
Distinguish conditional directions, joint mass, expectation and posterior probability.
Describe random quantities
Compute expectation and variance for discrete outcomes and distinguish a continuous density from probability mass.
Average outcomes before they happen
Can an expected payoff be an amount you never actually receive?
Read a density as probability per unit
Connect continuous probability, area, Gaussian scale and numerical integration.
Explain a decision under uncertainty
Build a fictional decision model with an explicit outcome space, conditional probabilities and numerical consequences. Compare alternatives using expected outcomes while explaining a downside that the average hides.
Build from lesson evidence, compare alternatives, write an artifact, and tackle unfamiliar numerical cases. Use the rubric with a teacher, peer or colleague to assess the quality of your reasoning.
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