Omikuji, priced: six fortunes, a 30.9% hit rate, and the two-大吉 card that pays nothing

One roll decides the card

An omikuji card looks like a scratch game — nine cells, scratch them, hope for three of a kind — but the engine decides the whole card with one number before any cell exists. The first draw from the round’s seed stream is an integer from 0 to 999,999. The six fortune tiers own consecutive ranges of that million, in the order below; if the roll lands in a tier’s range the card wins that tier, and if it lands in the remaining 691,000 the card pays nothing. Only then are the glyphs painted on.

Multipliers are total return per unit staked (×1.5 returns $15 on $10, a $5 profit). “Points” are percentage points of stake.

Read the right-hand column and the card inverts: the fortune everybody scratches for is the smallest contributor to the return, and the fortune nobody wants — 末吉, “blessing to come”, a 50-cent profit on a dollar — is the largest. That is not a quirk; it is how a 30.9% hit rate and a ×100 top prize fit inside the same 96.55%. Two thirds of the return is paid in ×1.5, ×2 and ×4 wins, which is where almost every session’s money actually comes back from.

What the edge costs

A 3.45% edge is a mid-priced card on this site — about a third more than Chinchiro or roulette, a little under the slots. Per 100 cards at $10 the expected cost is $34.50, which slots between Bingo ($33.40) and Tairyō (about $37) on the per-hundred table. Pace is yours: a card every ten seconds is 360 cards an hour, about $124 at $10 and $12.40 at $1.

Standard deviation per $1 card is 3.77 (second moment 15.1); over 100 cards ×√100 ×$10. The 大吉 chance is 1 − 0.999¹⁰⁰; the 中吉-or-better chance is 1 − 0.993¹⁰⁰.

The standard deviation is eleven times the expected loss, which is the shape of every top-heavy paytable: the session result is mostly noise, and the edge is what remains when the noise averages out. A 大吉 needs, on average, a thousand cards; half of all players who keep going will have seen one by card 693, and a quarter will still be waiting at card 1,386. At $10 a card those are $6,930 and $13,860 of turnover for a $1,000 prize — which is the ×100 stated in dollars rather than in multipliers.

The two-大吉 card

Here is the part the scratch format exists for. A card has nine cells and seven kinds of glyph — the six fortunes plus 凶 — and the paint rule caps every non-winning glyph at two copies. Nine cells across seven kinds with a cap of two means at least two kinds must appear twice. So a losing card is never a spread of singles: it always carries at least two pairs, and usually three.

From 2,000,000 cards through the engine’s paint rule; the pair minimum of two is exact by counting (9 cells, 7 kinds, cap 2).

Put the two tables together. A card that pays ×100 is one in a thousand. A card that shows two 大吉 and pays nothing is 46% of 69.1%, which is 31.8% of all cards — 318 of them for every real 大吉. The “so close” card is not close to anything: the roll that decides the card was made before the glyphs were, and the two 大吉 are filler chosen from whatever was still allowed. The same is true inside a win — about a quarter of winning cards also show a pair of a higher tier than the one they paid, 末吉 with two 大吉 looking on.

This is the honest version of a design every scratch card and most slots use, and the near-miss piece covers why it works on people: two-of-three reads as progress even when the third was never in play. The difference here is that the rule is printed. Three of a kind pays; two of a kind is the normal appearance of a card that paid nothing, and it would look the same if the top tier were a million to one.

The shrine and the card

At a shrine the box is private theology: nobody publishes how many 大吉 slips are in it, and the custom of tying a bad fortune to the pine exists because the ratio is not yours to know. The history piece tells that story. The card does the opposite — the ratios are six integers in a source file — and the only mystery left is the one every weighted draw has, which is which range the next roll lands in.