Track 10 · Ruin · lesson 2
Expected value, and its two failure modes
12 min
Track 10 · Ruin · lesson 2
12 min
Suppose someone offers you a coin that is slightly bent. It comes up heads 52 times in a hundred rather than 50. You win 100 on heads, lose 100 on tails, and you may play as often as you like.
The arithmetic takes ten seconds. Two throws in a hundred are yours, each worth 200 of swing, so the average throw pays you 4.
The arithmetic is not the hard part. Knowing when that number describes your life is the hard part, and it fails in two specific ways.
Expected value is the average result of a decision, taken over every outcome, each weighted by how likely it is.
Multiply each outcome by its probability. Add them up. That is the whole operation, and it is the correct way to compare two gambles whose odds you actually know.
What it gives you is the long-run average per attempt. What it does not give you is any statement at all about what happens on the attempt you are about to make.
The expected value of one throw of that bent coin is 4. You will never win 4. You will win 100 or lose 100, and the 4 is a description of a very large number of throws that mostly have not happened.
That is not pedantry. It is the difference between a number that describes you and a number that describes a spreadsheet.
Predict
Below is the same bet. The expected total climbs in a straight line. The band around it climbs too, but as the square root of the number of trials rather than in proportion to it, which is the only reason repetition helps at all.
Expected value
Hypothetical model, not a forecast
The average outcome of a repeated two-sided bet, and how widely the total can still land around it.
It assumes
It ignores
Keeping the odds and the stake exactly as they are, find how many trials it takes to push the chance the total is still negative below one in four.
| Trials | Expected total | One spread above | One spread below |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 1 | 4 | 104 | -95.92 |
| 2 | 8 | 149 | -133 |
| 3 | 12 | 185 | -161 |
| 4 | 16 | 216 | -184 |
| 5 | 20 | 243 | -203 |
| 6 | 24 | 269 | -221 |
| 7 | 28 | 292 | -236 |
| 8 | 32 | 315 | -251 |
| 9 | 36 | 336 | -264 |
| 10 | 40 | 356 | -276 |
| 11 | 44 | 375 | -287 |
| 12 | 48 | 394 | -298 |
| 13 | 52 | 412 | -308 |
| 14 | 56 | 430 | -318 |
| 15 | 60 | 447 | -327 |
| 16 | 64 | 464 | -336 |
| 17 | 68 | 480 | -344 |
| 18 | 72 | 496 | -352 |
| 19 | 76 | 512 | -360 |
| 20 | 80 | 527 | -367 |
Not there yet — keep moving the controls.
It takes somewhere around 285 trials before the chance of being behind drops under one in four. Roughly three hundred repetitions of a real, known, mathematically certain edge, and one time in four you are still down.
Now consider that most real decisions offer you five or six repetitions in a lifetime.
Expected value is a statement about a large number of trials. Its usefulness falls off sharply as the number of trials you will actually get falls towards one.
For a decision you will make hundreds of times — a pricing rule, an insurance book, a hiring process — the expectation is close to the point. For a decision you will make once, it is one input among several, and often not the important one.
Look at the assumptions rendered above the model. The last one says that nothing stops the run early, including running out of money.
That is not a small simplification. It is the entire difference between the average and your life, and it is the subject of the next lesson.
For now, notice the shape of the problem. Expected value averages across every possible sequence of results — including the sequences where you went to zero on throw seven and, in the arithmetic, carried on throwing anyway.
Check
The model reports one figure that gets less attention than it deserves: the win chance at which the bet is exactly fair.
It is the cleanest way to interrogate a proposition you cannot price. Rather than arguing about whether the chance of success is 30% or 45%, compute the chance at which the thing breaks even, and then ask a much easier question — is the true probability above or below that line?
People are poor at estimating probabilities and considerably better at comparisons. Turning an estimation problem into a comparison is usually the whole trick.