The 1% rule is a way to limit a chosen stake, not a mathematical guarantee of safety. Its first question is what the percentage refers to: the initial session allocation or the balance immediately before each wager. Those choices produce different balance paths. Neither removes the cost of a negative-return game, and neither creates an obligation to deposit enough money to fit a preferred percentage.
This article uses one declared independent-round model: accepted 2x targets win with probability 0.485 and otherwise lose. It is an educational assumption, not an observed universal return for Lucky Jet, Aviator or JetX. We distinguish a specified loss sequence, a sequence anywhere in a session, an administrative balance cutoff and a 50% loss from the initial amount. Calling all four “ruin” would hide the actual question.
Fixed initial percentage and current percentage
Starting with 500, a fixed initial 1% stake is 5 on every round. A 2x win brings the bank to 505, but the next stake stays 5. A loss brings it to 495, again with stake 5. Its percentage of the remaining bank changes. This is a fixed-unit model:100 initial stakes fit the starting amount, before considering payments, additional exposure or fees.
At 1% of the current balance, the same first win makes the next stake 5.05; the first loss makes it 4.95. Without rounding, a loss multiplies the balance by 0.99 and a 2x win by 1.01. In general a current fraction f gives factors 1-f and 1+f. A sequence of k losses leaves B(1-f)^k. A fixed initial fraction instead removes kf of the initial bank while those wagers can still be funded.
For example, four losses at current 1% remove 3.940399%, not exactly 4%. At current 5% they remove 18.549375%, not 20%. The fixed-initial versions would remove 4% and 20% respectively. Real minimum stakes or currency rounding can change either calculation. If a minimum forces a larger percentage, the actual exposure must be disclosed rather than keeping the 1% label.
A loss block is different from a run anywhere
Under this model q=0.515. A particular block of k future rounds all losing has probability q^k. Asking whether at least one run of k losses occurs anywhere in 100 rounds is another event. Its windows overlap; adding their probabilities or treating every possible window as independent gives the wrong answer. A longer run also contains shorter runs and multiple overlapping windows.
| Loss run | Specified block | At least once in 100 | Current 1% loss | Current 5% loss |
|---|---|---|---|---|
| 4 | 7.034430% | 98.180191% | 3.940399% | 18.549375% |
| 6 | 1.865707% | 60.285999% | 5.851985% | 26.490811% |
| 8 | 0.494832% | 20.568959% | 7.725531% | 33.657957% |
| 10 | 0.131242% | 5.726765% | 9.561792% | 40.126306% |
The 100-round probabilities are approximately 98.180191%, 60.285999%, 20.568959% and 5.726765% for lengths 4,6,8 and 10. They refer to at least one run of that length or longer, not exactly one run or a run beginning at a preselected position. The last two columns show the balance loss caused by that run under current fractions 1% and 5%. A run probability is not the probability of exhausting a whole session bank.
Previous outcomes do not change the next model probability under the stated independence assumption. That assumption is not established merely by calling an engine HMAC or by looking at a short history. Tracking your actual accepted stakes and balances remains useful, but the model does not promise that a long losing sequence is due to be followed by a win.
A reproducible 500-round fractional model
Now start each hypothetical path at 500 and use current fractions 1%, 2.5% or 5% for at most 500 rounds. Keep target 2x and p=0.485. There is no currency rounding, minimum stake, fee, replenishment or profit stop. A balance below 1 is an administrative cutoff after settlement, including the last round. It is not literal zero: with f<1, both balance factors are positive for any finite path.
Separately count whether the path ever falls below 250, half the initial balance. That event is not a complete loss and is not drawdown from a moving peak. The calculation can count its first crossing without claiming the session actually stopped there. These definitions must stay visible beside the numbers: changing the threshold, horizon or stake rule changes the result.
| Current fraction | Balance after 500 losses | P(below 1) | P(below 250) | Below 250 in 20000 simulations |
|---|---|---|---|---|
| 1.000000% | 3.285242 | 0% | 1.605707% | 323/20000 |
| 2.500000% | 0.001590 | 1.476803e-23% | 52.795747% | 10612/20000 |
| 5.000000% | 3.637246e-09 | 0.001099% | 87.119304% | 17513/20000 |
At 1%, even 500 losses leave 500×0.99^500≈3.285242, above 1. Thus the cutoff probability is exactly zero for this model and horizon. The probability of ever going below 250 is nevertheless about 1.605707%. At 2.5% and 5%, the below 1 probabilities are positive, while crossing below 250 is much more common. A zero administrative-cutoff probability is not a promise that a large loss cannot occur.
The table’s last column records a finite seeded simulation, not another definition of probability. We ran 20000 model sessions per fraction with Python seeds 20261023,20261024 and 20261025. The 50% loss counts were 323,10612 and 17513. None crossed below 1 in that sample, including fractions whose exact model probability is nonzero. No observed event is not proof that the event is impossible.
How the numerical checks work
The downloadable fractional_bankroll.py uses dynamic programming over round count and wins. A state’s balance is 500(1+f)^w(1-f)^(n-w). Barrier comparisons use exact integer products, while the larger probability calculation uses floating mass. Small examples can use exact rational probabilities. This avoids classifying a threshold by accumulated rounding in the main calculation.
Independent checks enumerated every path in 54 small configurations using fractions. The loss-run results were also checked with a separate exact recurrence for avoiding a run. The simulation rates agreed with DP within a two-sided Hoeffding absolute bound 0.013784867 for each individual event, confidence 99.9%, at 20000 sessions. That is about 1.378487 percentage points, not a joint confidence guarantee for every table cell.
Save the module in the same directory as the following Python example. All four language versions contain the same executable code and reproduce the same reference values. To add the optional simulation, runpython fractional_bankroll.py --sessions 20000. It uses the standard library and performs no network requests or real wagers.
# Download fractional_bankroll.py into this directory first.
# Educational2x model only, without currency rounding or minimum bets.
import json
import logging
from fractional_bankroll import reference_table
result = reference_table()
assert result["start"] == 500 and result["rounds"] == 500
assert result["fractions"][0]["below1"] == 0
assert result["fractions"][0]["allLossBalance"] > 1
logging.basicConfig(level=logging.INFO, format="%(message)s")
logging.info(json.dumps(result, indent=2))One percent is not fractional Kelly here
For 2x net odds b=1, Kelly’s unconstrained expression is(bp-q)/b=-0.03. Restricting the stake to nonnegative fractions selects zero. Half or quarter of that constrained choice is still zero. A positive 1% entertainment stake therefore is not fractional Kelly derived from a positive advantage. It is a chosen way of spending a limited amount despite negative expectation.
Smaller fractions can alter the distribution and rate of spending. They do not make a future recovery necessary or remove the cost of repeated accepted turnover. The expected-value guide distinguishes expected cost from a realised path. Do not use the low cutoff number as a reason to increase total exposure or play until a profitable result arrives.
Session limits are not single-win controls
A hypothetical stop after losing 100 from a 500 start concerns settled session net or a balance threshold 400. A profit target 150 concerns balance 650, not a single gross win of 150. A field named “stop if one win exceeds” answers a different question and includes different payments. Check the actual version’s accepted settings rather than assuming the same controls exist everywhere.
Stopping after a settled crossing can overshoot a loss boundary by the amount at risk. If a hard spending floor must be respected, assess the full possible loss before accepting another wager and include simultaneous positions. Reducing stakes or adding such stops would change the modelled tables above. A preset can support a decision without guaranteeing protection against every client, server or payment failure.
Keep the affordable amount fixed
If the available minimum does not fit your spending limit, there is no mathematical instruction to add money. You can choose no wager. The Bankroll Calculator shows fixed initial-unit examples and discloses its 0.10USD illustrative minimum; it does not calculate this full current-fraction distribution. The open module supplies that separate educational reference.
Partner actions remain direct registration choices without a compulsory demo step. They do not confirm that a named game implements these abstract stakes or thresholds. A percentage is useful only while its denominator, funding rule and stopping definition remain clear. Keep essential money outside the entertainment allocation and do not replenish a loss merely to preserve a numerical label.
