Probability Trees and Conditional Expectations

Probability tools for investment decisions under uncertainty: expected value and dispersion of a random outcome, probability trees with conditional expectations, and Bayes' formula for updating a probability when new information arrives.

1. 1. Expected value, variance and standard deviationExpected value is your probability-weighted forecast; variance and standard deviation measure how far the actual outcome is likely to land from that forecast.Video · 7 minFree14 min
2. 2. Conditional probability and probability treesA probability tree splits an uncertain outcome into scenarios and then into outcomes conditional on each scenario, so joint and total probabilities can be read straight off its branches.14 min
3. 3. Conditional expected values and variancesWork out the expected value inside each scenario, then weight those conditional expectations by the scenario probabilities: the result must match the overall forecast.Video · 6 min15 min
4. 4. Bayes' formula: updating a probability with new informationBayes' formula turns a prior probability into a posterior probability by multiplying it by how much more, or less, likely the new information is when the event is true.Video · 5 min15 min
5. 5. Bayes in practice: diffuse priors, missing inputs and screening testsExam Bayes problems usually hide one input: you may need to back it out with the total probability rule, use complements, or start from equal (diffuse) priors.14 min
6. 6. Integrated cases: a tree with model forecasts and a Bayes updateWhen the end points of a tree are forecasts from a fitted line, turn each branch value into a model output first, then use conditional expectations to roll the tree back, and let Bayes' formula revise the scenario weights when news arrives.15 min