Part V · Chapter 11 · Ruin · lesson 2

Expected value, and its two failure modes

12 min

Builds on Diversification stops working when you need it

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.

Failure mode one: the average is not an outcome

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

You have that 52-in-100 edge, even money, same stake every time. You play 100 times. Roughly what is the chance you are still behind at the end?

Under 1 in 20 — the edge is real and a hundred is a lot of throwsAbout 1 in 10About 1 in 3About 1 in 2 — a coin is a coin

Watching the spread swallow the edge

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

Every trial has exactly the same odds and is independent of the one before.
The stake never changes, whatever has happened up to that point.
The band around the total uses a normal approximation, which is rough at low trial counts.
Nothing interrupts the run, including the player running out of money.

It ignores

Ruin — this model happily continues past a balance of zero
Any change in bet size after a win or a loss
Correlation between trials, and odds that shift over time
Costs of playing, such as a commission on each trial

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.

Number of trials20

Expected result per trial: 4
Expected total: 80
Spread of one trial: 99.92
Spread of the total: 447
Chance the total is still negative: 0.429
Win chance that makes this fair: 50%

Expected total
One spread above
One spread below

A straight expected line rising from zero, inside a band that widens as the square root of the trial count and only slowly stops covering zero.
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.

Failure mode two: the calculation assumes you are still there

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

Which of these decisions is expected value best suited to?

Setting the excess on a policy an insurer will write two hundred thousand timesDeciding whether to put most of your savings into one private businessChoosing which of two job offers to acceptDeciding whether to take a bet that would end you one time in twenty

The breakeven probability, which is the useful outputOptional

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.

Worth remembering

Expected value is each outcome multiplied by its probability, summed — the average result per attempt over a long run.
The average is not an outcome: a bet with an expected value of 4 pays 100 or minus 100, never 4.
The spread of a total grows with the square root of the number of trials while the expected total grows in proportion, which is why repetition helps slowly.
Assume a small edge on an even-money bet and it can take roughly three hundred repetitions before being behind becomes unlikely.
Expected value assumes the run continues, including through the sequences where the money already reached zero.