Half Kelly Calculator
The formula
What the half kelly tells you
The plain odds behind half kelly: 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.
Half Kelly: the number, the method and the caveats
This page was written for one job: turning your own inputs into a defensible figure, with the working shown. There are plenty of pages about Half Kelly; almost none of them show the working. This one does: the formula, every input, a fully computed example, and a sensitivity analysis calculated live from this engine’s own defaults.
The useful way to hold Half Kelly is as a ratio with a story attached: p = fav ÷ poss; P(at least one hit in n trials) = 1 − (1−p)^n says how the parts combine, and the 40.00% figure above is simply that sentence written in numbers. Everything that follows unpacks one clause of the sentence at a time.
People rarely disagree about arithmetic; they disagree about assumptions that were never written down. Printing the substitution and the sensitivity ranking removes that category of disagreement for Half Kelly.
Nothing below is stock text pasted around a number: the worked example computes single-trial probability = 40.00% 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. Read the sections in order and you will be able to reproduce every figure by hand.
How Half Kelly is computed, line by line
One formula, published in full: p = fav ÷ poss; P(at least one hit in n trials) = 1 − (1−p)^n. Anything you can read, you can reproduce in a spreadsheet, and that reproducibility is the whole point of printing it rather than hiding it behind a button.
Favourable outcomes (example value 2) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. If you are unsure what belongs in this field, look at the printed steps: the substitution shows exactly how the engine uses it.
Possible outcomes (example value 5) is a plain measured quantity — the unit shown in the label is the unit the formula expects. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
Repetitions (independent) (example value 4) is a measured number in the unit named by the label; the formula does not convert for you. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
A fully worked Half Kelly example
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 endpoint of that chain is 40.00% of single-trial probability, sitting in the “Live odds — worth calculating the payoff” band.
- p = 2 ÷ 5 = 0.4000
- Odds against = (1−p)÷p = 1.50:1
- P(hit within 4 tries) = 1 − 0.6000^4 = 87.0 %
Odds against — 1.50 :1. If your inputs change, this is usually the first companion figure to move.
At least one hit in 4 — 87.0%. If your inputs change, this is usually the first companion figure to move.
Fair payout to break even — 2.50 :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.
How far Half Kelly can reasonably move
The honest version of any estimate is a corridor, and this one is cheap to produce: Half Kelly 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 | 50.00% |
| 90 % of the example value | 44.44% |
| example inputs | 40.00% |
| 110 % of the example value | 36.36% |
| 120 % of the example value | 33.33% |
Across 5 settings of Possible outcomes, the result spans 33.33% to 50.00%. 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 Half Kelly, ranked by effect
This list was computed for this specific engine just now: each numeric input was nudged ±10 % while every other field held its example value, and the single-trial probability was recalculated from scratch. It is the fastest honest answer to “what matters most in Half Kelly?” — and it is specific to these defaults, because sensitivity moves with the starting point.
- Possible outcomes: ten percent up gives 36.36%, ten percent down gives 44.44% — ≈11.1 % of the single-trial probability either way
- Favourable outcomes: ten percent up gives 44.00%, ten percent down gives 36.00% — ≈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 36.36% to 44.44%. Measure that input from a document rather than from memory, and treat Favourable outcomes as the second priority (≈10.0 % per ten percent). Everything below that can stay approximate, which is a licence to stop chasing data that would not change the decision.
What a Half Kelly of 40.00% means in practice
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. Before treating the band as fact, note its width: ten percent either way on Possible outcomes already spans 36.36% to 44.44%. The practical reading rules: read the band as a starting hypothesis, then test it: change the two dominant inputs by ten percent and see whether the verdict survives. Verdicts that survive stress are the ones worth acting on.
Set Half Kelly beside its neighbours and it usually plays a specific role: this figure establishes the size of something, and the neighbouring tools establish whether that size is acceptable. Skipping the first step makes the second one arbitrary.
Why Half Kelly results go wrong — and how to prevent it
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 Half Kelly 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.
Half Kelly verification checklist
- Note the date and the inputs beside the number so the figure can be audited later.
- Verify each input against its original source document, not memory.
- Check Possible outcomes before anything else: it is the measured dominant input, ≈11.1 % of Half Kelly per ten percent.
- Confirm the example values are plausible for your own situation before replacing them.
- Record the date and the inputs alongside the number so future-you knows what was assumed.
- Repeat the ±10 % test on the inputs that matter and see whether the decision — not just the number — still holds.
How to use the Half Kelly tool properly
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 Half Kelly 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. When in doubt, prefer the boring input: the interesting one is usually the one you misread.
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 (40.00% for the example values), and note the verdict band. If a result feels obviously right, that is the moment to check the inputs hardest.
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. When in doubt, prefer the boring input: the interesting one is usually the one you misread.
Step 4 — compute a corridor, not a point. Three runs — worst, likely, best — turn Half Kelly from a snapshot into a range. Plan against the pessimistic end and treat the optimistic end as the upside story. Say the answer out loud before you save it; if it sounds wrong, the model did not change, your understanding did.
Step 5 — write the one-line memo. “Single-trial probability = 40.00% 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. One decimal place more than your data supports is one decimal place of fiction.
The real-world conversations Half Kelly settles
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. When two sources disagree, it is almost never the arithmetic that differs.
The companion outputs — Odds against and At least one hit in 4 — exist for the second stage of that screen: the headline sizes the prize, while the companions describe the trade-off you are accepting to get it. Estimates improve most when the worst case is written down in ink.
There is a personal-administration use people overlook: periodic self-checking. Half Kelly quietly drifts as life changes — rates move, bodies change, costs compound — and re-running it a few times a year catches drift while correction is still cheap. Write the assumptions in the same file as the result, and future arguments take minutes instead of meetings.
The questions that come up most about Half Kelly
What affects Half Kelly the most?
Measured on this engine, Possible outcomes is the dominant lever: moving it ±10 % shifts the single-trial probability across 36.36% to 44.44% — 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?
By the numbers, Possible outcomes: a ten-percent change in it moves the result ≈11.1 %, versus ≈10.0 % for Favourable outcomes. Spend your measuring effort in that order — precision on the weaker lever buys almost nothing.
What is a typical Half Kelly?
On the worked example above Half Kelly comes out at 40.00%. Treat that as one point on a distribution rather than a norm: the bands, the companion outputs and the sensitivity ranking tell you how wide the plausible range really is.
How do I calculate Half Kelly 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 Half Kelly than any number of calculator runs.
What happens to Half Kelly if possible outcomes changes by 10 %?
The engine was re-run with that field moved by ten percent while everything else stayed fixed: the single-trial probability goes across 36.36% to 44.44%, a swing of roughly 11.1 % around the 40.00% baseline. That ratio is the practical definition of how much this input matters.
Is the Half Kelly result of 40.00% good or bad?
Good and bad are comparisons, so the honest test is three questions: where does your own Half Kelly sit against the 40.00% reference above, how far would a plausible input error move it (≈11.1 % per ten percent on Possible outcomes), and what does the alternative option compute to?
Can Half Kelly be negative or empty?
Yes, and that is informative rather than broken: a negative Half Kelly usually means the inputs encode an outflow or a shortfall, and the verdict band above labels exactly that situation instead of hiding it.
How often should I check Half Kelly?
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. Half Kelly is a snapshot, not a subscription; stale snapshots are the most common silent error in Miscellaneous & Fun planning.
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.
Do I need an account to use this Half Kelly calculator?
No, and no data leaves the tab. The engine runs in the page itself, which is why the worked example, the bands and the sensitivity ranking all respond instantly to your keystrokes.
The vocabulary of Half Kelly
- Odds against
- 1.50 :1 on the example inputs — a companion figure printed beside the headline of Half Kelly so that no result is read in isolation.
- At least one hit in 4
- 87.0% on the example inputs — a companion figure printed beside the headline of Half Kelly so that no result is read in isolation.
- Fair payout to break even
- 2.50 :1 on the example inputs — a companion figure printed beside the headline of Half Kelly so that no result is read in isolation.
- Single-trial probability
- The primary output of this tool: 40.00% 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 calculation written out in full so that nothing on this page has to be taken on trust.
- 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 Half Kelly moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
The provenance of this Half Kelly calculation
There is no black box on this page — the engine evaluates 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. Specialist software layers extra coefficients on top of the same algebra; none are hidden here, and that transparency is what makes a disagreement traceable to a specific input instead of a black box.
This page is an estimation aid, not professional advice. The design rule is simple: two people entering identical figures must get identical answers. The promise to the reader is narrow and testable: same inputs, same output, no silent rounding, and no figure presented without the formula that produced it.
After the number: what to do with it
Re-run the calculation with your own figures, stress the two most sensitive inputs by ±10 %, and only then act on the result.
The audience here is people who need one trustworthy number before their next decision. Everything you need to defend the result — formula, substitution, sensitivity, source — is printed above, so the only remaining variable is the honesty of your inputs.