What happened
In one million 3/10 draws, 1-2-3 appeared 8,209 times and 2-6-9 appeared 8,294 times. Each had the same expectation, 1,000,000/120 ≈ 8,333.333.
The experiment uses a reduced model deliberately. A million 6/49 draws would have expected count only about 0.0715 for a specified set, usually leaving both comparison counts at zero. The 3/10 model makes the sampling variation visible while retaining the same set-counting logic.
Method and benchmark
Enumerate C(10,3)=120 subsets exactly, then sample 1,000,000 uniform 3/10 draws. Compare two combinations chosen before simulation. This is explicitly a reduced model, not a 6/49 jackpot experiment.
All C(10,3)=120 combinations are enumerated exactly before sampling. The two comparison sets are fixed in the program, not selected because of their final counts. The complete frequency plot avoids hiding less attractive observations.
Python 3.12.14 · NumPy 2.3.5 · NumPy PCG64 · seed 20260925. Each experiment starts a separate stream. Code and all datasets are linked below.
What this does not establish
Equal probability is established by counting. Finite sample counts will differ. A million 6/49 trials usually gives zero observations of a specified combination, so it would be a poor demonstration.
Counting proves the model’s equality: every three-label set has the same number of ordered selections. Simulation illustrates the variability around that equality. It cannot replace the argument, and the observed difference is not a persistent advantage for either set.
These are original educational simulations prepared for RoomForChance. They are not historical lottery records, a physical-machine audit, an external peer review or evidence of a prediction advantage.
Inspect the data
Download ordered-versus-irregular.csv · All results and metadata (JSON) · Download the complete Python program · Download figure-generation code
| combination | observed | expected |
|---|---|---|
| 1-2-3 | 8,209 | 8,333.33333 |
| 1-2-4 | 8,351 | 8,333.33333 |
| 1-2-5 | 8,290 | 8,333.33333 |
| 1-2-6 | 8,355 | 8,333.33333 |
| 1-2-7 | 8,221 | 8,333.33333 |
| 1-2-8 | 8,285 | 8,333.33333 |
| 1-2-9 | 8,365 | 8,333.33333 |
| 1-2-10 | 8,135 | 8,333.33333 |
| 1-3-4 | 8,407 | 8,333.33333 |
| 1-3-5 | 8,444 | 8,333.33333 |
| 1-3-6 | 8,313 | 8,333.33333 |
| 1-3-7 | 8,264 | 8,333.33333 |
| 1-3-8 | 8,365 | 8,333.33333 |
| 1-3-9 | 8,245 | 8,333.33333 |
| 1-3-10 | 8,350 | 8,333.33333 |
| 1-4-5 | 8,321 | 8,333.33333 |
| 1-4-6 | 8,428 | 8,333.33333 |
| 1-4-7 | 8,258 | 8,333.33333 |
| 1-4-8 | 8,464 | 8,333.33333 |
| 1-4-9 | 8,212 | 8,333.33333 |
| 1-4-10 | 8,258 | 8,333.33333 |
| 1-5-6 | 8,371 | 8,333.33333 |
| 1-5-7 | 8,289 | 8,333.33333 |
| 1-5-8 | 8,349 | 8,333.33333 |
| 1-5-9 | 8,226 | 8,333.33333 |
| 1-5-10 | 8,293 | 8,333.33333 |
| 1-6-7 | 8,183 | 8,333.33333 |
| 1-6-8 | 8,278 | 8,333.33333 |
| 1-6-9 | 8,256 | 8,333.33333 |
| 1-6-10 | 8,378 | 8,333.33333 |
| 1-7-8 | 8,281 | 8,333.33333 |
| 1-7-9 | 8,469 | 8,333.33333 |
| 1-7-10 | 8,362 | 8,333.33333 |
| 1-8-9 | 8,348 | 8,333.33333 |
| 1-8-10 | 8,518 | 8,333.33333 |
| 1-9-10 | 8,381 | 8,333.33333 |
| 2-3-4 | 8,277 | 8,333.33333 |
| 2-3-5 | 8,336 | 8,333.33333 |
| 2-3-6 | 8,356 | 8,333.33333 |
| 2-3-7 | 8,297 | 8,333.33333 |
| 2-3-8 | 8,324 | 8,333.33333 |
| 2-3-9 | 8,222 | 8,333.33333 |
| 2-3-10 | 8,289 | 8,333.33333 |
| 2-4-5 | 8,429 | 8,333.33333 |
| 2-4-6 | 8,422 | 8,333.33333 |
| 2-4-7 | 8,205 | 8,333.33333 |
| 2-4-8 | 8,314 | 8,333.33333 |
| 2-4-9 | 8,502 | 8,333.33333 |
| 2-4-10 | 8,283 | 8,333.33333 |
| 2-5-6 | 8,424 | 8,333.33333 |
| 2-5-7 | 8,256 | 8,333.33333 |
| 2-5-8 | 8,305 | 8,333.33333 |
| 2-5-9 | 8,250 | 8,333.33333 |
| 2-5-10 | 8,235 | 8,333.33333 |
| 2-6-7 | 8,354 | 8,333.33333 |
| 2-6-8 | 8,282 | 8,333.33333 |
| 2-6-9 | 8,294 | 8,333.33333 |
| 2-6-10 | 8,443 | 8,333.33333 |
| 2-7-8 | 8,355 | 8,333.33333 |
| 2-7-9 | 8,297 | 8,333.33333 |
| 2-7-10 | 8,265 | 8,333.33333 |
| 2-8-9 | 8,189 | 8,333.33333 |
| 2-8-10 | 8,287 | 8,333.33333 |
| 2-9-10 | 8,318 | 8,333.33333 |
| 3-4-5 | 8,467 | 8,333.33333 |
| 3-4-6 | 8,371 | 8,333.33333 |
| 3-4-7 | 8,439 | 8,333.33333 |
| 3-4-8 | 8,302 | 8,333.33333 |
| 3-4-9 | 8,286 | 8,333.33333 |
| 3-4-10 | 8,357 | 8,333.33333 |
| 3-5-6 | 8,283 | 8,333.33333 |
| 3-5-7 | 8,219 | 8,333.33333 |
| 3-5-8 | 8,250 | 8,333.33333 |
| 3-5-9 | 8,362 | 8,333.33333 |
| 3-5-10 | 8,526 | 8,333.33333 |
| 3-6-7 | 8,187 | 8,333.33333 |
| 3-6-8 | 8,307 | 8,333.33333 |
| 3-6-9 | 8,346 | 8,333.33333 |
| 3-6-10 | 8,356 | 8,333.33333 |
| 3-7-8 | 8,428 | 8,333.33333 |
| 3-7-9 | 8,353 | 8,333.33333 |
| 3-7-10 | 8,390 | 8,333.33333 |
| 3-8-9 | 8,451 | 8,333.33333 |
| 3-8-10 | 8,378 | 8,333.33333 |
| 3-9-10 | 8,350 | 8,333.33333 |
| 4-5-6 | 8,374 | 8,333.33333 |
| 4-5-7 | 8,327 | 8,333.33333 |
| 4-5-8 | 8,224 | 8,333.33333 |
| 4-5-9 | 8,278 | 8,333.33333 |
| 4-5-10 | 8,418 | 8,333.33333 |
| 4-6-7 | 8,312 | 8,333.33333 |
| 4-6-8 | 8,317 | 8,333.33333 |
| 4-6-9 | 8,336 | 8,333.33333 |
| 4-6-10 | 8,362 | 8,333.33333 |
| 4-7-8 | 8,283 | 8,333.33333 |
| 4-7-9 | 8,465 | 8,333.33333 |
| 4-7-10 | 8,420 | 8,333.33333 |
| 4-8-9 | 8,400 | 8,333.33333 |
| 4-8-10 | 8,283 | 8,333.33333 |
| 4-9-10 | 8,531 | 8,333.33333 |
| 5-6-7 | 8,320 | 8,333.33333 |
| 5-6-8 | 8,316 | 8,333.33333 |
| 5-6-9 | 8,232 | 8,333.33333 |
| 5-6-10 | 8,468 | 8,333.33333 |
| 5-7-8 | 8,417 | 8,333.33333 |
| 5-7-9 | 8,305 | 8,333.33333 |
| 5-7-10 | 8,316 | 8,333.33333 |
| 5-8-9 | 8,439 | 8,333.33333 |
| 5-8-10 | 8,193 | 8,333.33333 |
| 5-9-10 | 8,254 | 8,333.33333 |
| 6-7-8 | 8,373 | 8,333.33333 |
| 6-7-9 | 8,341 | 8,333.33333 |
| 6-7-10 | 8,389 | 8,333.33333 |
| 6-8-9 | 8,447 | 8,333.33333 |
| 6-8-10 | 8,391 | 8,333.33333 |
| 6-9-10 | 8,451 | 8,333.33333 |
| 7-8-9 | 8,263 | 8,333.33333 |
| 7-8-10 | 8,266 | 8,333.33333 |
| 7-9-10 | 8,385 | 8,333.33333 |
| 8-9-10 | 8,364 | 8,333.33333 |
Reproduce the experiment
Download the Python program to an empty working folder. Use the recorded environment for an exact replay. The program runs all eleven studies and creates a lab-data folder containing the result files. The largest study performs ten million draws; allow time for it to finish.
python -m pip install numpy==2.3.5
python run_experiments.pyExpected CSV SHA-256: a955135f7c19ab61819622e749a01c4ef11027c1256780f08d59b98de8682d4e. A matching seed alone is insufficient if you change the implementation or call sequence. The CSV files use CC BY 4.0; the original code uses the MIT license included with the downloads.
Read the reasoning
How Many Lottery Combinations Are There? The C(n,k) Formula
Derive the lottery combination formula, work through C(49,6), and learn when separate pools or ordered digits require a different count.
Combinations vs Permutations: Does Order Matter?
Distinguish unordered lottery sets from ordered digit games, with replacement rules and examples that prevent common counting errors.
Why Humans Do Not Choose Numbers Randomly
Explore the difference between human selection and uniform sampling, with research on number preferences and a measurable coin-sequence example.
Lottery Probability Explained: Count the Outcomes First
A complete starting point for lottery odds: sample spaces, combinations, separate pools, exact matches and the limits of simulation.
How to Generate Random Numbers Without Repetition
Learn two fair methods for distinct random numbers, why sorting is safe, and why filtering patterns changes the distribution.
Is 1-2-3-4-5-6 as Likely as Any Other Lottery Combination?
Yes under a uniform 6/49 model. Understand why a specific ordered-looking line differs from the broad category of all consecutive sets.
Technical references
NumPy PCG64 documentation describes the generator family. Harvard Stat 110 provides university-level probability background. The model-specific derivation is linked above.