About the math¶

This page explains, briefly, why every other page on this site is labelled "descriptive only". The TL;DR is: lottery draws are independent and identically distributed (i.i.d.) uniform random samples. Past draws contain no information about future draws. The full list of myths debunked is at the bottom.

The gambler's fallacy¶

The most common mistake: thinking that because a number has not been drawn for a long time, it is due. This is the gambler's fallacy.

In a fair draw, on every single draw, the probability of any specific ball being chosen is the same as on every previous draw. The fact that ball 16 has not appeared in 180 draws is a fact about the past; 180 happens to be inside the expected range of gap lengths for any given ball in a 6/55 lottery.

The complementary mistake is the hot-hand fallacy: thinking a number that has appeared frequently recently will continue to do so. Same reason: independence.

How long should a gap be?¶

Roughly geometric with mean equal to pool_size / picks_per_draw.

• Power 6/55: 55/6 = ~9 draws per appearance on average.
• Mega 6/45: 45/6 = 7.5.
• Max 5/35: 35/5 = 7 (per draw -- and there are two draws per day).

But that's the mean -- gaps can easily be 5x the mean and still be unremarkable. A 50-draw gap in 6/55 has a probability of about (49/55)^50 ~= 0.4% per ball, but with 55 balls being checked every draw, you'll see one such gap surprisingly often.

Pairs, triplets and lift¶

Pages on this site display the "most common pairs" and "top triplets". With C(55, 2) = 1485 possible pairs and ~1000 draws, even the most common pair is usually only 1-2 standard deviations above the mean. This is expected noise, not signal. We surface a chi-square check on the pair lift so you can see for yourself that no pair is statistically significant once you Bonferroni-correct.

Why "balanced" picks are not better¶

Many guides suggest picking numbers that match the historical distribution (e.g. sum near the mean, equal odd/even split). These patterns are more common in the historical record -- but only because there are more combinations matching them. Each individual ticket has the same chance of winning the jackpot.

The one tiny edge "balanced" picks have: if your ticket follows the shape most other players use, you are more likely to share the prize with someone else. So if you really want to maximise expected payout conditional on winning, pick an unusual combination. But again, this does not change the chance of winning at all -- only the size of your share if you do.

Day of week and weather¶

Some sites overlay "by Tuesday" or "by month" charts of the drawn numbers. We include those too, but they're labelled clearly: a fair draw is invariant to the day. The seasonality you do see comes from player behaviour (more tickets sold during big jackpots and around holidays), not from the balls.

Why we still show this stuff¶

For the same reason coin-flip histograms are in every intro stats textbook: descriptive statistics on random data are educational. They let you feel what i.i.d. uniform looks like, and how big the "natural" deviations really are. They make the gambler's fallacy intuitive once you've stared at the residuals.

Myths and their corrections¶

Digit-pick games (Max 3D, Max 3D Pro)¶

The 3-digit games work differently from the classic ball-pick products: each draw produces 20 three-digit numbers (000–999) and the player's ticket must match one of them exactly (or, in some tiers, a suffix).

Every position is independent¶

A 3-digit number like 037 is a tuple of three independent uniform draws over {0, …, 9}. The hundreds digit, tens digit, and units digit each have an equal 1-in-10 chance, regardless of what the other positions show.

The per-position frequency charts on the Digits page are a direct check: with ~1000 draws and 10 digits per position, the expected count per digit is (1000 × 20) / 10 = 2000. A chi-square test on the pooled digit counts is included on the page.

Gap lengths¶

For digit-pick games the gap formula is pool / draws_per_day:

• Max 3D & Max 3D+ : face down on the same days, but each day draws 20 three-digit numbers. Pool = 1000, picks = 20 → expected gap 50 draws (~17 weeks).
• Max 3D Pro: same pool (1000) and same picks (20), so expected gap is also 50 draws (~17 weeks).

Number structure¶

The Numbers page breaks down the frequency of structural patterns: palindromes (121, 344), all-same (777, 000), ascending runs (123, 789), descending runs (987, 321), and pairs (122, 505). These are descriptive — they show how many three-digit combinations belong to each pattern class, not whether one class is "luckier".

Repeated numbers¶

Hypergeometric math says that if you draw 20 numbers from a bag of 1000 without replacement, the probability that at least one number from yesterday's draw repeats today is:

1 - C(980, 20) / C(1000, 20) ≈ 34%

The actual repeat rate on the Numbers page should be close to this (sampling noise aside). Seeing exactly the same number on consecutive days is not a "signal" — it is expected roughly once every three draws.

Hot/cold digits¶

Digits are not "hot" or "cold" in any predictive sense. A gap of 100+ draws for a specific digit (expected gap: 50) is uncommon but not suspicious, and the hot/cold lists on the Digits page are purely descriptive summaries of what has happened so far.

Myths about digit-pick games (additions)¶

For the source literature, see Clotfelter & Cook (1991), Terrell (1994), and the Wikipedia articles on the gambler's fallacy and the hot-hand fallacy.