Part V · Chapter 12 · Noise · lesson 5

Same decisions, different luck

13 min

Builds on Diversification stops working when you need it and The bet you must not take twice

Two people make the same hundred decisions. Same information, same reasoning, same sizing, same order. One ends up ahead and one ends up behind.

Nothing needs to have gone wrong for that to happen. It is the ordinary behaviour of a process with any randomness in it, and the whole of this lesson is an attempt to make that fact concrete enough to be uncomfortable.

A decision with an edge is not a decision that wins. It is a decision that wins more often than it loses, over a number of repetitions large enough for the difference to show through the noise.

The number of repetitions required is the part nobody has an intuition for. It is usually far larger than the number of decisions of that kind a person makes in a working life, which means a career is often too short a sample to distinguish a genuine edge from a run of ordinary luck.

A hundred good decisions

Take a decision that is genuinely favourable: it works 55 times in 100, it pays the same amount it costs, and each instance is independent of the last. That is a real edge. It is also a larger edge than most people ever actually have.

Run it a hundred times and look at the band, not the line.

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 repetition is independent and carries exactly the same odds as the last.
The amount at stake is identical on every repetition, whatever happened before.
The spread around the total uses a normal approximation, which is a simplification.
Nothing stops the run early, including the account reaching zero.

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

Leaving the odds alone, find the number of repetitions at which the chance of still being behind falls under 5%.

Number of trials100

Expected result per trial: 10
Expected total: 1,000
Spread of one trial: 99.5
Spread of the total: 995
Chance the total is still negative: 0.157
Win chance that makes this fair: 50%

Expected total
One deviation above
One deviation below

A straight expected line rising from zero, with a band around it that widens the further out it goes.
Period	Expected total	One deviation above	One deviation below
0	0	0	0
1	10	109	-89.5
2	20	161	-121
3	30	202	-142
4	40	239	-159
5	50	272	-172
6	60	304	-184
7	70	333	-193
8	80	361	-201
9	90	388	-208
10	100	415	-215
11	110	440	-220
12	120	465	-225
13	130	489	-229
14	140	512	-232
15	150	535	-235
16	160	558	-238
17	170	580	-240
18	180	602	-242
19	190	624	-244
20	200	645	-245
21	210	666	-246
22	220	687	-247
23	230	707	-247
24	240	727	-247
25	250	747	-247
26	260	767	-247
27	270	787	-247
28	280	806	-246
29	290	826	-246
30	300	845	-245
31	310	864	-244
32	320	883	-243
33	330	902	-242
34	340	920	-240
35	350	939	-239
36	360	957	-237
37	370	975	-235
38	380	993	-233
39	390	1k	-231
40	400	1k	-229
41	410	1k	-227
42	420	1.1k	-225
43	430	1.1k	-222
44	440	1.1k	-220
45	450	1.1k	-217
46	460	1.1k	-215
47	470	1.2k	-212
48	480	1.2k	-209
49	490	1.2k	-206
50	500	1.2k	-204
51	510	1.2k	-201
52	520	1.2k	-197
53	530	1.3k	-194
54	540	1.3k	-191
55	550	1.3k	-188
56	560	1.3k	-185
57	570	1.3k	-181
58	580	1.3k	-178
59	590	1.4k	-174
60	600	1.4k	-171
61	610	1.4k	-167
62	620	1.4k	-163
63	630	1.4k	-160
64	640	1.4k	-156
65	650	1.5k	-152
66	660	1.5k	-148
67	670	1.5k	-144
68	680	1.5k	-140
69	690	1.5k	-136
70	700	1.5k	-132
71	710	1.5k	-128
72	720	1.6k	-124
73	730	1.6k	-120
74	740	1.6k	-116
75	750	1.6k	-112
76	760	1.6k	-107
77	770	1.6k	-103
78	780	1.7k	-98.75
79	790	1.7k	-94.36
80	800	1.7k	-89.94
81	810	1.7k	-85.49
82	820	1.7k	-81
83	830	1.7k	-76.48
84	840	1.8k	-71.92
85	850	1.8k	-67.33
86	860	1.8k	-62.71
87	870	1.8k	-58.06
88	880	1.8k	-53.38
89	890	1.8k	-48.67
90	900	1.8k	-43.93
91	910	1.9k	-39.16
92	920	1.9k	-34.36
93	930	1.9k	-29.53
94	940	1.9k	-24.68
95	950	1.9k	-19.79
96	960	1.9k	-14.88
97	970	1.9k	-9.95
98	980	2k	-4.99
99	990	2k	0
100	1k	2k	5.01

Not there yet — keep moving the controls.

At a hundred repetitions the expected total is well above zero and the band still comfortably includes it. Someone running this correctly a hundred times finishes behind roughly 16% of the time — not because they did anything wrong, but because that is what a 55/45 edge does over a hundred tries.

Getting the chance of being behind under one in twenty takes somewhere around 270 repetitions.

Predict

Given that, what does a five-year losing record actually tell you about a strategy?

That the strategy has no edge — five years is plenty of evidenceVery little, unless the number of independent decisions in those five years was largeThat the person running it made errors, since a real edge would have shownNothing at all — outcomes never contain information about strategies

The draw that ends the run

The model above has a convenient feature that reality does not: it happily continues past a balance of zero. Add the constraint that you stop when the money runs out and the same decisions produce a different distribution of lives.

Probability of ruin

Hypothetical model, not a forecast

The chance a bankroll reaches zero over a run of even-money bets, even when the bets are in your favour.

It assumes

Every bet stakes the same amount and pays even money when it wins.
The bankroll is counted in whole stakes, and any remainder is rounded away.
Betting stops permanently the moment the bankroll reaches zero.
Each bet is independent, with identical odds on every single one.

It ignores

Bet sizing that scales with the bankroll, which changes everything
Any way to add money to the bankroll mid-run
A target at which the player would stop and walk away
Payouts other than even money, and any commission on winnings

Keeping the same favourable odds and the same bankroll, find a stake at which the chance of reaching zero inside the run goes above one in three.

Bankroll1,000

Staked on each bet100

The same amount every bet. The bankroll is counted in units of this.

Bankroll, counted in bets: 10
Edge on each bet: 10%
Expected result per bet: 10
Chance of ruin inside this run: 0.119
Chance of surviving this run: 0.881
Chance of ruin if the betting never stops: 0.134
Expected bankroll at the end, in bets: 28.47
Expected bankroll at the end: 2,847

Still solvent
Reached zero

Two curves from opposite corners: survival falling away from one, ruin climbing from zero, both flattening out.
Period	Still solvent	Reached zero
0	1	0
1	1	0
2	1	0
3	1	0
4	1	0
5	1	0
6	1	0
7	1	0
8	1	0
9	1	0
10	1	0
11	1	0
12	0.999	0.001
13	0.999	0.001
14	0.997	0.003
15	0.997	0.003
16	0.996	0.004
17	0.996	0.004
18	0.993	0.007
19	0.993	0.007
20	0.991	0.009
21	0.991	0.009
22	0.988	0.012
23	0.988	0.012
24	0.986	0.014
25	0.986	0.014
26	0.983	0.017
27	0.983	0.017
28	0.98	0.02
29	0.98	0.02
30	0.977	0.023
31	0.977	0.023
32	0.974	0.026
33	0.974	0.026
34	0.971	0.029
35	0.971	0.029
36	0.968	0.032
37	0.968	0.032
38	0.965	0.035
39	0.965	0.035
40	0.963	0.037
41	0.963	0.037
42	0.96	0.04
43	0.96	0.04
44	0.957	0.043
45	0.957	0.043
46	0.955	0.045
47	0.955	0.045
48	0.952	0.048
49	0.952	0.048
50	0.95	0.05
51	0.95	0.05
52	0.948	0.052
53	0.948	0.052
54	0.945	0.055
55	0.945	0.055
56	0.943	0.057
57	0.943	0.057
58	0.941	0.059
59	0.941	0.059
60	0.939	0.061
61	0.939	0.061
62	0.937	0.063
63	0.937	0.063
64	0.935	0.065
65	0.935	0.065
66	0.933	0.067
67	0.933	0.067
68	0.932	0.068
69	0.932	0.068
70	0.93	0.07
71	0.93	0.07
72	0.928	0.072
73	0.928	0.072
74	0.927	0.073
75	0.927	0.073
76	0.925	0.075
77	0.925	0.075
78	0.923	0.077
79	0.923	0.077
80	0.922	0.078
81	0.922	0.078
82	0.921	0.079
83	0.921	0.079
84	0.919	0.081
85	0.919	0.081
86	0.918	0.082
87	0.918	0.082
88	0.917	0.083
89	0.917	0.083
90	0.915	0.085
91	0.915	0.085
92	0.914	0.086
93	0.914	0.086
94	0.913	0.087
95	0.913	0.087
96	0.912	0.088
97	0.912	0.088
98	0.911	0.089
99	0.911	0.089
100	0.91	0.09
101	0.91	0.09
102	0.909	0.091
103	0.909	0.091
104	0.908	0.092
105	0.908	0.092
106	0.907	0.093
107	0.907	0.093
108	0.906	0.094
109	0.906	0.094
110	0.905	0.095
111	0.905	0.095
112	0.904	0.096
113	0.904	0.096
114	0.903	0.097
115	0.903	0.097
116	0.902	0.098
117	0.902	0.098
118	0.901	0.099
119	0.901	0.099
120	0.9	0.1
121	0.9	0.1
122	0.9	0.1
123	0.9	0.1
124	0.899	0.101
125	0.899	0.101
126	0.898	0.102
127	0.898	0.102
128	0.897	0.103
129	0.897	0.103
130	0.897	0.103
131	0.897	0.103
132	0.896	0.104
133	0.896	0.104
134	0.895	0.105
135	0.895	0.105
136	0.895	0.105
137	0.895	0.105
138	0.894	0.106
139	0.894	0.106
140	0.893	0.107
141	0.893	0.107
142	0.893	0.107
143	0.893	0.107
144	0.892	0.108
145	0.892	0.108
146	0.892	0.108
147	0.892	0.108
148	0.891	0.109
149	0.891	0.109
150	0.891	0.109
151	0.891	0.109
152	0.89	0.11
153	0.89	0.11
154	0.89	0.11
155	0.89	0.11
156	0.889	0.111
157	0.889	0.111
158	0.889	0.111
159	0.889	0.111
160	0.888	0.112
161	0.888	0.112
162	0.888	0.112
163	0.888	0.112
164	0.887	0.113
165	0.887	0.113
166	0.887	0.113
167	0.887	0.113
168	0.886	0.114
169	0.886	0.114
170	0.886	0.114
171	0.886	0.114
172	0.886	0.114
173	0.886	0.114
174	0.885	0.115
175	0.885	0.115
176	0.885	0.115
177	0.885	0.115
178	0.884	0.116
179	0.884	0.116
180	0.884	0.116
181	0.884	0.116
182	0.884	0.116
183	0.884	0.116
184	0.883	0.117
185	0.883	0.117
186	0.883	0.117
187	0.883	0.117
188	0.883	0.117
189	0.883	0.117
190	0.882	0.118
191	0.882	0.118
192	0.882	0.118
193	0.882	0.118
194	0.882	0.118
195	0.882	0.118
196	0.881	0.119
197	0.881	0.119
198	0.881	0.119
199	0.881	0.119
200	0.881	0.119

Not there yet — keep moving the controls.

The odds never changed. The edge is the same favourable edge throughout. What changed is how many consecutive bad draws the position could absorb before the run ended, and that is a decision about size rather than about direction.

Two people making identical decisions at different sizes are not running the same strategy at different intensities. They are running two strategies, and one of them has an absorbing state in it.

What would you do

Two colleagues have reached the same conclusion about the same opportunity, with the same reasoning and the same confidence. It is genuinely favourable. They differ only in how much of what they have they are prepared to commit, and each has to decide alone.

Optimising for: Being able to keep making the same decisions in ten years, rather than maximising the expected total.
You cannot know: How many consecutive bad outcomes this particular opportunity can produce
Whether the odds they have estimated are the odds that actually apply
Whether they will need the money for something else within the period

Commit an amount that four consecutive bad outcomes would wipe out.Commit an amount that twenty-five consecutive bad outcomes would wipe out.Commit an amount too small to matter either way.

What this means for reading anyone's record

Three things follow, and they are the reason this track sits before the track on building a business.

A track record is a sample, and usually a small one. Before treating someone's results as evidence about their ability, count the independent decisions. It is nearly always a smaller number than the years suggest.

Identical decisions produce a distribution, not a result. When you see one outcome, you are seeing one draw from a range you cannot observe. The person in front of you cannot see it either.

Survival is a separate variable from expectation. Two people with the same view and the same edge can end in very different places purely through sizing, and the one who sized aggressively will look more skilled for as long as the run lasts.

Worth remembering

A genuine edge produces losing stretches at a predictable rate, so a losing run is not evidence of a bad process.
The number of repetitions needed for an edge to show through the noise is usually far larger than a career contains.
Calendar time is not the unit — independent decisions are, and most financial decisions repeat slowly.
Adding an absorbing state at zero changes the outcome distribution even when the odds are unchanged.
Two people with identical views and different sizing are running different strategies, and one of them can end.