St. Petersburg Paradox Calculator
The formula
What the st. petersburg paradox tells you
The plain odds behind st. petersburg paradox: single-event probability, the fair-price payout it implies, and how repetition moves the chance of at least one success.
Calculation: p = fav ÷ poss; P(at least one hit in n trials) = 1 − (1−p)^n. Enter your inputs and every substitution is shown below the result.
Reading the result
The fair-payout line is the decision column — any offered price above it carries positive expected value, below it funds the house.
Limitations and best practice
Independence is the assumption; card draws, weather and streaks correlate, which shrinks the repetition column’s real value.
This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Frequently asked questions
Do odds “even out”?
What changes with repetition?
Source & methodology
Classical probability (Laplace) for equally-likely outcomes.
St. Petersburg Paradox explained: formula, worked example and honest limits
You are here because you need a specific number, and you want to see how it was produced before you rely on it. Searchers land on a St. Petersburg Paradox page for three reasons — to plan, to compare options, and to check that somebody else’s arithmetic is honest. This tool serves all three, and everything after the calculator explains the model instead of hiding it.
In plain terms, St. Petersburg Paradox answers the question "Single-trial probability?". The measurement matters because it converts a situation that feels complicated — several inputs pulling in different directions — into a single number you can act on. Within Miscellaneous & Fun, Gambling & Probability is the context where this figure is quoted most often, and professionals treat it as a first-pass screen rather than a verdict.
Fun numbers are still numbers. St. Petersburg Paradox computes p = fav ÷ poss; P(at least one hit in n trials) = 1 − (1−p)^n honestly, and the 28.57% output is exactly the sort of thing that settles a group chat — with a real formula underneath instead of a joke answer.
Nothing below is stock text pasted around a number: the worked example computes single-trial probability = 28.57% from this tool’s own defaults, and the sensitivity section ranks the inputs that move the result — Possible outcomes first, at ≈11.1 % of the headline for a ten-percent error either way in that input. Work down the page and by the end you could re-derive the result in a spreadsheet from memory.
Every input in St. Petersburg Paradox, and what it does to the answer
The engine implements a single stated model: p = fav ÷ poss; P(at least one hit in n trials) = 1 − (1−p)^n. There is no hidden coefficient and no login-gated precision — the formula printed is the formula executed, which is also what makes the result citable.
Favourable outcomes (example value 2) is a quantity rather than a category — precision here is cheap and worth having. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
Possible outcomes (example value 7) is a measured number in the unit named by the label; the formula does not convert for you. The example value is there to demonstrate the mechanics, not to describe your situation — overwrite it.
Repetitions (independent) (example value 3) is a quantity rather than a category — precision here is cheap and worth having. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
St. Petersburg Paradox with real numbers, step by step
Rather than describe the calculation, here it is end to end — every substitution in order, starting from the inputs above. Starting figures: Favourable outcomes, Possible outcomes, Repetitions (independent). The computed single-trial probability comes out at 28.57%, which the interpretation band labels “Live odds — worth calculating the payoff”.
- p = 2 ÷ 7 = 0.2857
- Odds against = (1−p)÷p = 2.50:1
- P(hit within 3 tries) = 1 − 0.7143^3 = 63.6 %
Odds against — 2.50 :1. Compare it against the main result of 28.57% before drawing conclusions.
At least one hit in 3 — 63.6%. Read next to the headline, this is the figure that shows the shape of the trade-off.
Fair payout to break even — 3.50 :1. This is the number experienced users quote when the headline alone would mislead.
Keep this worked example open while you enter your own figures: matching the shape of the substitution is the fastest way to spot a unit or convention mistake.
Scenario table: St. Petersburg Paradox at ±10 % and ±20 % of the key input
The honest version of any estimate is a corridor, and this one is cheap to produce: St. Petersburg Paradox was recomputed at 5 points along the dominant input, with every other field left untouched.
| Possible outcomes setting | Single-trial probability |
|---|---|
| 80 % of the example value | 35.71% |
| 90 % of the example value | 31.75% |
| example inputs | 28.57% |
| 110 % of the example value | 25.97% |
| 120 % of the example value | 23.81% |
Across 5 settings of Possible outcomes, the result spans 23.81% to 35.71%. A decision that only works at one row of this table is a decision waiting for an input to move. Reproduce it in the form above by editing Possible outcomes alone; the substitution steps will follow your figures exactly.
The levers behind St. Petersburg Paradox, ranked by effect
Nobody should have to guess which input matters. The engine was re-run once per field below — ten percent up, ten percent down, everything else frozen — and the results ranked by how far they moved the single-trial probability. It is the fastest honest answer to “what matters most in St. Petersburg Paradox?” — and it is specific to these defaults, because sensitivity moves with the starting point.
- Possible outcomes +10 % → St. Petersburg Paradox = 25.97% · −10 % → 31.75% (the single-trial probability shifts ≈11.1 %)
- Favourable outcomes: ten percent up gives 31.43%, ten percent down gives 25.71% — ≈10.0 % of the single-trial probability either way
Possible outcomes is the dominant lever here: ten percent either way moves the result by about 11.1 %, from 25.97% to 31.75%. That is where a careless figure does the most damage, so check it against a source, and treat Favourable outcomes as the second priority (≈10.0 % per ten percent). The remaining fields can sit at safe defaults without materially changing the answer — knowing which is which is what separates a careful calculation from an anxious one.
Interpreting the St. Petersburg Paradox verdict
A number on its own is trivia; a number against a range is intelligence. The verdict attached to the default calculation — “Live odds — worth calculating the payoff” — is one band on a ladder your own figure may sit somewhere else on. The sensitivity list shows how far a modest input error moves you along that ladder: for Possible outcomes, ±10 % covers 25.97% to 31.75%. The practical reading rules: ask what would have to change to move you one band. If the answer is “a small, plausible input difference”, the band itself is not the finding — the uncertainty is.
Where St. Petersburg Paradox sits next to similar measures: this tool answers “how much is the St. Petersburg Paradox”, while the related calculators answer “what do I do about it”. Reading the result alone tends to over-weight a single snapshot; pairing it with the companion tools above turns the number into a decision.
How people get St. Petersburg Paradox wrong
Independence is the assumption; card draws, weather and streaks correlate, which shrinks the repetition column’s real value. This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Stopping at the first plausible answer. If the result surprises you, the productive question is which input would have to be wrong for the surprise to disappear — then go check that input.
This model has its own particular trap: Possible outcomes carries ≈11.1 % of the response per ten percent of error, so effort spent elsewhere is largely decorative.
Another is comparing across contexts: a St. Petersburg Paradox computed with one set of conventions should never be lined up against a figure computed with another. Check the inputs behind both before ranking them.
A quick check for any St. Petersburg Paradox figure
- Compare the output against one published benchmark or a colleague’s independent estimate.
- Check that every field the formula needs actually arrived from a document, not from recollection.
- Re-verify Possible outcomes against a source document — ten percent of error there moves the headline by ≈11.1 %.
- State, in one sentence, which convention each rate, unit or period in the form is using.
- Confirm the example values are plausible for your own situation before replacing them.
- Confirm the answer against one independent method or published benchmark for St. Petersburg Paradox.
A routine for decision-grade St. Petersburg Paradox
The tool does the arithmetic; the routine below is what makes the arithmetic trustworthy. Five steps, a minute total — and afterwards you will know not just what the number is, but which assumption produced it.
Step 1 — start from evidence. The inputs behind St. Petersburg Paradox are favourable outcomes, possible outcomes, repetitions (independent). Write them down with their source next to each one; if you cannot name where a figure came from, that is the input most likely to be wrong. Suspiciously tidy answers deserve the same scrutiny as untidy ones.
Step 2 — enter the conservative case first. Resist the flattering scenario. Type the inputs you would defend in front of a sceptical reader, read the Single-trial probability (28.57% for the example values), and note the verdict band. When in doubt, prefer the boring input: the interesting one is usually the one you misread.
Step 3 — play with the inputs on purpose. Ten percent up, ten percent down, one field at a time. Within a few minutes you will know this calculation better than most people who quote its results. A result worth acting on survives having its two shakiest inputs halved.
Step 4 — run the scenario trio. Expected, pessimistic, optimistic. The gap between pessimistic and expected is the number to plan against; the optimistic result is useful only as a ceiling. The test of a good estimate is not that it feels right but that it changes what you do.
Step 5 — write the one-line memo. “Single-trial probability = 28.57% on this date, from these inputs.” That line is the entire deliverable; everything else on this page is how you earned the right to write it. If the verdict changes with a ten-percent nudge, report the nudge alongside the number.
From calculation to decision: where St. Petersburg Paradox earns its keep
In Gambling & Probability the figure computed here is usually the first one somebody asks to see. Having it — with the formula behind it — changes your position in the conversation from participant to reference. Rounding early is the quietest way to introduce a permanent error.
Professionals in Gambling & Probability use this style of calculation as a screening filter before spending real money or effort: a quick St. Petersburg Paradox check eliminates weak options in minutes, and the survivors earn the deeper, slower analysis. A figure nobody can reproduce is a figure nobody will act on twice.
Nothing here is stored or transmitted, so the habit of checking is free: open the page, re-enter current figures, compare with the number you saved last time. When two sources disagree, it is almost never the arithmetic that differs.
St. Petersburg Paradox — frequently asked questions
What affects St. Petersburg Paradox the most?
Measured on this engine, Possible outcomes is the dominant lever: moving it ±10 % shifts the single-trial probability across 25.97% to 31.75% — about 11.1 % of the headline, with Favourable outcomes next at ≈10.0 %. The ranked list in the sensitivity section is computed live, not asserted from a template.
Which matters more: favourable outcomes or possible outcomes?
For St. Petersburg Paradox, Possible outcomes wins the head-to-head (≈11.1 % against ≈10.0 % per ten-percent change). Knowing which input dominates is the difference between careful work and merely slow work.
What is a typical St. Petersburg Paradox?
With the example inputs on this page the result is 28.57%, labelled “Live odds — worth calculating the payoff”. “Typical” varies with situation, which is why every result here ships with its verdict band and its sensitivity list: they turn one figure into a range you can locate yourself inside.
How do I calculate St. Petersburg Paradox by hand?
Three steps: write the formula (p = fav ÷ poss; P(at least one hit in n trials) = 1 − (1−p)^n), insert your values, and simplify. The worked example section shows exactly that chain for the default inputs; replicating it once on paper teaches you more about St. Petersburg Paradox than any number of calculator runs.
What happens to St. Petersburg Paradox if possible outcomes changes by 10 %?
Try it in the form above and the printed substitution will follow you; the pre-computed answer for the defaults is 25.97% to 31.75% across the ten-percent probe, i.e. ≈11.1 % sensitivity on the single-trial probability.
Is the St. Petersburg Paradox result of 28.57% good or bad?
The tool answers “what”, not “whether”: 28.57% carries the verdict “Live odds — worth calculating the payoff” for the default case, which is a starting interpretation rather than a verdict on your situation. Re-run with your own figures and compare the bands.
How often should I check St. Petersburg Paradox?
Whenever an input changes by more than a few percent, and otherwise on the rhythm of your decisions — before a purchase, a review, a renewal. St. Petersburg Paradox is a snapshot, not a subscription; stale snapshots are the most common silent error in Miscellaneous & Fun planning.
What affects St. Petersburg Paradox the most?
Sensitivity depends on where you start, but the inputs Favourable outcomes and Possible outcomes typically move St. Petersburg Paradox further than the rest because they enter the formula multiplicatively. If you can only measure two things well, measure those and hold the others at conservative defaults.
Do odds “even out”?
Each trial is memoryless — the law of large numbers guarantees averages over the long haul, not corrections soon.
What changes with repetition?
The at-least-one line compounds (1−(1−p)^n) — 3 % once per try becomes 1−0.97^30 ≈ 60 % across 30 tries.
Can this tool see the numbers I type?
It cannot. There is no submission step: the calculation happens in your browser and the result disappears when you close the page. Treat the fields above as a scratchpad rather than a form.
What the labels on this page mean
- Odds against
- 2.50 :1 in the default St. Petersburg Paradox calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- At least one hit in 3
- 63.6% in the default St. Petersburg Paradox calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Fair payout to break even
- 3.50 :1 for the St. Petersburg Paradox worked case; it sits next to the main result because it carries context the headline leaves out.
- Single-trial probability
- The primary output of this tool: 28.57% for the default inputs, computed exactly from p = fav ÷ poss; P(at least one hit in n trials) = 1 − (1−p)^n.
- Worked example
- The step-by-step substitution printed above, so the arithmetic can be re-derived by hand or in a spreadsheet.
- Verdict band
- A named zone on the range of plausible results; the example inputs land in “Live odds — worth calculating the payoff”, and your inputs may not.
- Sensitivity (±10 %)
- How far the St. Petersburg Paradox moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Method, sources and what this tool cannot tell you
The model is published arithmetic implemented exactly as written: p = fav ÷ poss; P(at least one hit in n trials) = 1 − (1−p)^n. Source of record: Classical probability (Laplace) for equally-likely outcomes. Any other implementation of this model should return the same figure for the same inputs. If one does not, the difference is in its assumptions — which this page prints and that one probably does not.
This page is an estimation aid, not professional advice. The design rule is simple: two people entering identical figures must get identical answers. The sensitivity table above is part of that promise rather than a bonus: it lets you see how much of the result depends on how carefully you measured.
Turning this St. Petersburg Paradox into a decision
Save the number together with its inputs and today’s date; a St. Petersburg Paradox figure without context is unusable the moment an input changes.
This page was written for people who need one trustworthy number before their next decision, and the promise is that nothing above required trust. Bookmark the tool, re-run it whenever an input moves by more than a few percent, and keep the date with the number.