Define the decision perimeter
State whether profit and loss are measured against initial cost, whether expenses are included, when cash becomes available and whether the listed events exhaust the possible outcomes. If a table omits the probability of delay or an additional contribution, a neat formula cannot repair an incomplete problem statement. It is better to show an unknown range than assign a convenient zero.
An unpleasant asymmetry
Option A produces +1,000 conditional roubles with probability 90% and −12,000 with probability 10%. Its expected value is −300: 900 of expected positive value does not cover 1,200 of expected loss. Option B produces +3,000 with probability 40% and −1,000 with probability 60%. Its expected value is +600. All probabilities are invented for teaching, not estimated from market data.
Choosing only by win frequency would prefer A and miss the size of the rare loss. Choosing only by expectation is also incomplete. If a loss of 1,000 would disrupt a family’s mandatory payment, positive expectation for B does not make it acceptable. Separate money for obligations first, as in the [emergency-fund guide](/guides/emergency-fund/), then discuss the residual risk.
Sensitivity instead of one attractive number
Suppose B’s probability of a positive outcome is not known exactly. Without additional costs, the break-even probability is 25%: 4,000 multiplied by the success probability must cover the possible 1,000 loss. At 20%, expected value is already −200. Replace the argument over one “+600” figure with two questions: why should probability exceed one quarter, and how reliable is the source of that estimate?
Offer three scenarios: cautious, base and favourable. Keep the result definition and horizon identical in all three. Do not change probability, outcome size and expense composition at the same time without explanation; otherwise it is impossible to see which assumption drove the conclusion. The [probabilistic forecast log](/guides/forecast-calibration/) is more useful than memory of successful predictions.
Money and utility are not the same
University presentations distinguish expected utility, where the utility of outcomes is averaged rather than the monetary amount itself. Risk attitude can make expected utility differ from the utility of expected money. This explains why giving up some monetary expectation for predictability is not necessarily an arithmetic error.[1]
Do not turn this idea into a pseudo-precise personal coefficient. An educational analysis can state the maximum tolerable loss, liquidity reserve, horizon and prohibition on debt. Two people can correctly calculate the same table and reach different conclusions because their obligations differ. The calculation structures the choice but does not replace a universal recommendation.
Separate luck from a good process
Save the table before the event. Afterward, check not only the outcome but whether the realised scenario was included, whether the original information was available and whether probabilities were changed retrospectively. One loss does not disprove a positive-expectation decision, and one win does not justify a negative-expectation one. With dependent repetitions and limited capital, even individually attractive opportunities may be impossible to sustain. Expected value must therefore be accompanied by loss capacity and the ability to keep meeting obligations.
The value of additional information
Before choosing, ask whether it is worth paying for a more precise probability estimate. First calculate the best expected result without new information, then calculate the result with a specifically described signal quality and all possible errors. The difference is an upper bound on the economic interest in the refinement within the model. It does not justify any research price and does not imply that a paid forecast is informative.
Do not put perfect knowledge of the future into the table as a practically available service. It can serve only as a theoretical comparison boundary. If new information cannot change the admissible action because of capital or time constraints, its monetary value for this problem may be small. The term [expected value](/glossary/matematicheskoe-ozhidanie/) keeps the calculation separate from impressions about attractiveness.
Decision tree: branch probability is not leaf probability
A single-decision table does not describe sequential choice. A tree distinguishes a chance node, where outcomes are averaged, from a decision node, where an available action is chosen. Compute from leaves to root. The probability of a complete path is the product of conditional probabilities along it; add mutually exclusive complete paths. MIT’s educational material discusses expected utility; the following tree and figures are independently constructed mathematical examples.[REV]
Continue artificial project B: success pays 3,000, failure −1,000, with initial success probability 0.4. Rejecting gives 0, so without new information the best expected result is 600. A research test now gives a positive signal with probability 0.75 if success occurs and 0.25 if failure occurs. The four complete leaf probabilities are: success and positive 0.30; failure and positive 0.15; success and negative 0.10; failure and negative 0.45. They sum to one.
The positive-signal probability is 0.45; conditional success after it is 0.30 / 0.45 = 2/3. Expected project value on the positive branch is 1,666.67. After a negative signal, success probability is 0.10 / 0.55 = 2/11 and expected value is −272.73. At this node choose rejection with result 0 rather than launching anyway. The value of the available choice before paying for information is 0.45 × 1,666.67 + 0.55 × 0 = 750, rounding only the final totals. If the test costs 100 for either signal, root value is 650. Information added 150 before its price and 50 after it, relative to the original 600.
Perfect information would give 0.4 × 3,000 + 0.6 × 0 = 1,200; its value is 600. That is an upper bound for refinement in this model, not the return of an available forecast. If rejection after the signal is impossible, the test creates no value here: the same project runs for either answer and the fee only lowers expectation. If obligations prohibit even a possible 1,000 loss, change the admissible action set first rather than altering probabilities for convenience.
When a tree cannot honestly be filled with probabilities
A poorly structured problem differs from a fully specified game: the outcome list may be incomplete, information may conflict, participants may have different goals and the decision deadline may be short. Recognising a familiar pattern and mentally running one plausible scenario can help choose what to investigate first, but does not validate probabilities. Do not assign 0.7 merely because an analyst quickly recognised a pattern.
A limited protocol is: record the objective and deadline; mark unknowns that could change admissibility; propose two materially different scenarios; trace the sequence of cash requirements; and name the observation that would make the analogy fail. Then collect discriminating information, choose a reversible action with a pre-limited loss or decline. Without defensible probabilities, show a result range rather than presenting it as a complete chance tree. Preserved records before the outcome are more useful than an expert’s confidence after it. A physiological reaction in a card experiment does not establish an ability to identify a profitable stock intuitively.
The St Petersburg game: infinite sum and finite utility
Consider a mathematical game, not advice to play: a fair coin is tossed until the first heads; if it appears on toss N, the payoff is X = 2ᴺ. The probability N = n is 2⁻ⁿ. Each monetary-expectation term 2⁻ⁿ × 2ⁿ equals one. The sum over all n therefore diverges: E[X] = ∞. This is a property of an infinite model; real cash, time and credit are not infinite.
If the maximum payoff is 2¹⁰ and exactly that maximum is paid for N ≥ 10, the first nine terms contribute 9 and the tail contributes 2¹⁰ × P(N ≥ 10) = 2. Expected capped payoff is 11. Do not confuse this cap with a different game that pays nothing after the tenth toss; its tail and sum differ. Changing the cap changes the object being calculated.
With logarithmic utility, u(X) = ln X, E[u(X)] = ln2 × Σn/2ⁿ = 2ln2 ≈ 1.3863, a finite amount. This does not imply a universal “reasonable ticket price”: utility of total wealth after an entry fee c is E[ln(W − c + X)], not E[ln X] − c. For W > 0 and 0 ≤ c ≤ W the expression is defined; its upper bound ln(W + 1) + Nln2 has finite expectation. Borrowing, inability to meet obligations and non-logarithmic preferences require a different setup. Logarithm is an example of diminishing marginal utility, not a diagnosis of a person’s risk attitude.
A normal distribution has a specified probability shape and finite moments; it is not a necessary condition for an expectation to exist. A heavy tail can also have a finite mean. What matters is payoff together with probability and the speed at which the tail decays, not the “normal/non-normal” label. Independence of repetitions is another assumption. Do not declare all financial returns normal or all their means meaningless at once.
The analogy to growth stocks is limited: a rapidly growing future payoff can dominate valuation if its probability and discounting restrain growth too weakly. Limit horizon, resources and competition rather than declaring today’s price justified by infinite expectation. A real company is not a coin with known probability. The framework can teach choice, expectation and information value without importing card stories or bets or promising repeatable gains.
Statistical experience: a winning series versus skill
Define a win as positive excess over a preselected comparable benchmark after expenses for one non-overlapping period. In an artificial null model, win probability is 1/2, periods are independent and the rule and performer are selected before observation. The number of wins K in ten periods is binomial. The probability of at least eight wins is (45 + 10 + 1) / 1024 = 0.0546875, about 5.47%. This is not the probability that the performer “has no skill”; it is the probability of data this extreme or more under the stated null model.
Even ten wins out of ten give 1/1024, but after screening 100 independent performers the probability of finding at least one such winner is 1 − (1023/1024)¹⁰⁰ ≈ 9.31%. Real funds share factors and overlapping positions, violating independence. The figure therefore cannot be mechanically applied to markets. Similar results may reflect one common risk rather than 100 independent tests.
Continue skill assessment with a pre-registered holdout series, unchanged success definition, excess magnitude, uncertainty range, turnover and severe losses. If gains occur only in a rising market, test whether common exposure explains them; if they disappear after costs, distinguish predictive information from accessible net performance. A 5% threshold is not a boundary between talent and luck. Failure to reject the null with ten periods may mean little information, not absence of skill. Report the full search and dependence; a famous winner’s biography does not replace this test.
The winning-group problem adds selection after the result. If a researcher groups winners under a school, industry or style only after seeing the table, the group itself was selected. It cannot be evaluated as though it had been the sole predeclared object. Shared style can explain common risk, common data and common positions rather than independent confirmation of each participant’s skill.
A separate protocol is: preserve all 100 eligible performers before the test, including those that later close; record the style assignment date, group-switch rules, benchmark and costs. Form groups using information available then, not final performance. First calculate each group’s excess over control, then test common exposure. If ten participants held nearly the same portfolio, one good year is not ten independent experiments. Renaming groups creates no new information.
Counterfactual uncertainty is the unobserved result of the same group under a different common shock. Additional periods and an explicitly justified dependence model can study it, but a famous biography cannot remove it. Closed performers, failed groups and all tried classifications remain in the report. A frozen-group holdout period can add evidence of process stability; it still does not prove innate talent or guarantee future net results. The test concerns a limited process hypothesis, not a winner legend.
Sources and verification boundaries
[1] [Lecture 20: Uncertainty](https://ocw.mit.edu/courses/14-01-principles-of-microeconomics-fall-2023/mit14_01_f23_lec20.pdf). Expected monetary value and expected utility.
[REV] MIT OpenCourseWare, “Lecture 20: Uncertainty,” source rm-expected-utility. It supports distinguishing monetary expectation from utility. The tree, infinite-game derivation and binomial examples are independent mathematical examples with explicit assumptions, not empirical claims.