Apply the rules of probability, expected value and Bayes' theorem to investment questions.
A probability is a number from 0 to 1. The probabilities of all possible outcomes sum to 1.
Events are independent if knowing one tells you nothing about the other. Many investing errors come from assuming independence that is not there: mortgages across a country, or stocks in a crash.
The probability-weighted average outcome. A trade with a 60% chance of gaining $100 and a 40% chance of losing $120 has an expected value of 0.6 x 100 minus 0.4 x 120 = +$12. A high win rate does not make a strategy profitable: a 90% chance of gaining $10 with a 10% chance of losing $200 has an expected value of minus $11.
Two bets with the same expected value can differ greatly in spread. Position size should reflect both, which is the idea behind the position-sizing rules in Manage Your Risk.
How to update a belief when new evidence arrives:
Suppose 10% of fund managers are truly skilled. A skilled manager beats the market in 70% of years; an unskilled one in 50%. A manager beat the market this year. What is the chance she is skilled?
One good year moves the probability only from 10% to 13.5%. After five winning years in a row it reaches about 37%: (0.10 x 0.7^5) / (0.10 x 0.7^5 + 0.90 x 0.5^5) = 0.0168 / 0.0449. Evidence accumulates slowly when luck alone produces many winners.
The starting probability (10% here) matters as much as the evidence. Ignoring it is base-rate neglect, the error behind chasing last year's top fund.
Averages of many independent outcomes settle toward the expected value. A small edge repeated many times is reliable; a single bet is not. The law does not say that losses will be made up, only that they are diluted.
In a loan book, 2% of borrowers will default. A model flags 90% of future defaulters and also flags 15% of borrowers who will repay. What share of flagged borrowers will actually default?
A model that catches 90% of defaulters still produces flags that are wrong nine times in ten, because defaults are rare. The base rate dominates.
Two bonds each have a 5% chance of default. If defaults were independent, the chance both default would be 0.05 x 0.05 = 0.25%. In a recession, defaults cluster, and the joint probability can be several times higher. The mortgage-backed securities of 2007 were priced as if defaults across regions were nearly independent; they were not.
The number of ways to pick k successes from n trials is n! / (k! x (n minus k)!). There are 45 ways for a manager to beat the market in exactly 8 of 10 years. With a 50% chance each year, the probability of exactly 8 wins is 45 / 1,024 = 4.4%; of 8 or more, (45 + 10 + 1) / 1,024 = 5.5%, the figure in the practice mission.
For a repeated bet that pays even money with a win probability p, the fraction of capital that maximizes long-run growth is 2p minus 1 (more generally p minus q / b, where q = 1 minus p and b is the payoff per dollar risked). With p = 0.6, Kelly says bet 20%. Because estimates of p are uncertain and full Kelly produces brutal drawdowns, practitioners use a fraction of it, often half.
If the probability a stock beats earnings is 0.5, the probability its guidance is raised is 0.4 and the probability of both is 0.2, the probability of at least one is 0.5 + 0.4 minus 0.2 = 0.7. Forgetting to subtract the overlap double-counts it.
In a spreadsheet, simulate 1,000 fund managers with no skill (each year a 50% chance of beating the market) over 10 years using RAND. Count how many beat the market in at least 8 of the 10 years. Compare with the theoretical figure of about 5.5%.
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