LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
The law of total expectation writes one average as a weighted sum of smaller averages. If a population splits cleanly into groups, the overall mean is each group's mean times the chance of that group. The identity is just accounting. What fails in markets, the lecture says, is the hope that those pieces settle down if you wait. The groups' averages move. A trading record split into wins and losses can be decomposed the same way, and it still may not converge.
Why it matters
An unconditional mean hides which piece moved. The height picture is a toy: an overall mean they simply call 5 foot 5, and one cell of 5 foot 1 with probability 0.16. A continuous version integrates against a density. The lecture says "mass function" there. The object in the integral is a density. Either way, the mean compresses the whole space.
Conditioning is often easier. Average height among one group, times the share of that group, plus the same for the other group, returns the same 5 foot 5 if the split is a real partition. A partition covers the whole space and the pieces do not overlap. The union is everything. The intersection of two different pieces is empty.
The idea in plain words
In a fixed population, more samples pull a conditional mean toward the mean of that population. That is the law of large numbers, and it needs a distribution that is not changing. The lecture's counterexample is a school in 1902 and a school in 2026. Average height for the same labeled group need not match. Place, nutrition, and the century moved the distribution. So the unconditional mean moved too. You can still decompose both years and ask which piece changed: the group mean, or the share of the group. Stock returns from the early 2000s versus returns now are the financial version of that sentence. They are companies, not repeated draws from one machine.
A trade log uses the same identity. Wins and losses partition the trades. Expected P&L is the average win times the chance of a win, plus the average loss times the chance of a loss. The lecture says those four numbers are usually unstable. Finer partitions are allowed: low, medium, and high volatility, then macro or political conditions, then win versus loss inside those. As long as each cut is a partition, the law still adds up.
Edge, here, means the pieces stay reasonably stable: a positive mean and a spread that does not end the account. Height in one workplace will not jump tomorrow the way returns can. Over 10 to 100 years, even height moves. Finance moves sooner. In a fixed-distribution lab, more trades would converge. Outside it, you are not owed that. Walk-forward tests are the tool they name, with the bias that the test is historical. They liked a remark that this kind of empirical work is data mining. You can keep going only if the thing you found is structural.
How to use it
Pick a partition you could have assigned at the time, such as win or loss. Compute each bucket's average and its weight, then compare an earlier window with a later one. A change in the overall mean should show up in a bucket mean, a weight, or both.
Do not read a flat run of winning trades as a fixed mean. Finance can reprice the distribution before the sample is large. Stability across regimes is the evidence they want.
What usually goes wrong
The identity is an adding-up rule, not a promise that averages converge. Convergence needs a distribution that stays put. The two schools separate those ideas.
A win-loss split is the only cut, so a change in volatility gets blamed on "the strategy stopped working." The lecture wants the finer partitions first, so you can see which slice moved.
A backtest that mined the split gets called the structure. The talk agrees the test is historical, and it only claims you can continue if the thing you found is actually structural. It does not show an example that passed.
Questions people ask
What is the height formula doing?
It is a toy. Call the overall mean 5 foot 5. It equals the male conditional mean times the male share, plus the female conditional mean times the female share, if those two groups partition the sample. The 5 foot 1 figure is one cell, probability 0.16, not a census.
Do trading averages converge if you take more trades?
Only in the lecture's fixed-distribution lab. In the market, the win mean, the loss mean, and the two probabilities can move. More trades do not force them onto a constant.
Why compare 1902 with 2026?
To show a labeled conditional mean, height given a group, changing because the world changed. The lecture uses that as the picture for why an old return average is not the current one.