Hi-Lo, priced: the exact odds behind every card and every button

What the engine actually deals

Hi-Lo asks one question thirteen ways. A card is on the table; the next card will be higher or lower; call it, and a correct call multiplies the pot and hands you the same question about a new card. There is no shoe to track, no third-card rule, no dealer. That makes it one of the few casino games whose entire price list fits on a single page.

Ranks run Ace low to King high, one through thirteen. Because every card is an independent uniform draw, nothing that has already been dealt changes what comes next — no rank is “used up”, and a run of five low cards does not make a high one due. The history strip along the top of the board is a record, not a forecast.

That shared rank is the first thing worth noticing. On any card from 2 to Queen the two buttons together cover fourteen of the thirteen ranks — 107.69% — because a tie pays on whichever side you took. The house does not keep pushes here, and it does not need to: the price of both buttons is set as if the tie belonged to you, and the edge is taken somewhere else entirely.

Every call, priced

There are only twelve distinct calls in Hi-Lo, not twenty-six. A button is defined entirely by how many of the thirteen ranks win it, and every button on the board is one of the twelve sizes below.

Chance is the win-set size over 13. Fair odds is 13 ÷ win-set size, the figure hiloOdds() returns. The button shows the fair odds times the 1% return, floored to two decimals — a display, never a hair above what the ticket pays; from the second call of a run the same button shows the fair odds alone (×1.08, ×1.85 …) because the return has already been applied to the product.

The label is the only place two decimals appear. What the ticket accumulates is the exact fraction — 13 ÷ 12, 13 ÷ 7 — and the two-decimal figure on the button is a floor of that for display. It is safe to read the label as “at least this”.

One rounding, one edge

A Hi-Lo run does not pay per call. It accumulates. Each correct call multiplies a running product by that call’s exact odds, and the payout you can bank is that product with the house’s return applied, floored once at four decimals. The return is charged once for the whole run, not once per call, and so is the rounding.

House edge is the expected loss as a share of the stake for a run of exactly one correct call followed by a cash-out: 1 − chance × paid multiplier. The paid column is floor4(13 ÷ k × 0.99). The four-decimal floor costs at most 0.0038 points, on the ×1.8385 of a 7.

The two edge columns are the whole point, and they barely move. A configured return of 99% comes out as a house edge of 1.0000% on ten of the twelve calls and 1.0038% at worst, because a single four-decimal floor on a number between 1 and 13 can never take more than a hundredth of a percent. The near-certain ×1.07 and the ×12.87 long shot cost the same 1%. Which button you press decides how the result is distributed, not what it costs.

Averaged over the thirteen equally likely cards. The spread between the rows is half a thousandth of a point — for practical purposes there is no cheaper side.

What riding the ladder costs

The cash-out decision is the game. Every additional call multiplies the product by an exact fraction, the return is applied once at the end, and the floor is applied once at the end too. So a long run costs the same 1% as a short one; what changes is how rarely you arrive.

A run taken entirely on the 92.31% button. “Fair multiplier” is (13 ÷ 12) to the power of the call count — what the run would pay with no return applied. Paid is floor4(fair × 0.99). The largest rounding loss in the table is 0.0077 points at three calls; it shrinks as the multiplier grows because a fixed four-decimal floor is a smaller share of a bigger number.

Ten correct calls at even-ish odds is about one attempt in 488.

Both tables say the same thing: the price of a run is one percent of the stake, however long the run and whichever rungs it climbs. Doubling on the safe button takes ten correct calls and arrives a little under half the time; reaching ×100 takes sixty and happens about once in 122 attempts. The odds compound honestly, and nothing else compounds with them. That is what a single rounding at the end buys, and it is the reason the verifier has to reproduce that one floor exactly and no others.

Skipping, and what it does not do

Hi-Lo has a third button. While a run is live you can skip the card on the table and deal another one without calling it — the pot stands, the multiplier stands, and nothing is risked.

It is genuinely free, and since version 3 it buys even less than it used to. Skipping replaces one card with another drawn from the same thirteen ranks, and every card now offers the same 1% price on both buttons, so there is nothing cheaper to wait for. What a skip still does is let you choose the shape of the next call — a near-certain small step or a long shot — and cap how many times you can do that at fifty-two.

What changed in version 3

The first edition of this article priced the previous rule, and the difference is worth recording because a ticket from before the change still settles under it. Version 2 stored each won step floored to two decimals and then floored the finished multiplier to two decimals again. Two roundings, both downward, on every rung: a single call cost between 1.00% and 2.15% depending on how ugly its decimals were, and ten straight ×1.08 calls carried a 4.33% edge. Version 3 keeps the steps exact and rounds once, at four decimals, at the end. Every response from the engine carries the ticket’s version number, and the fairness panel shows which rule a given round was paid under.

The honest summary of Hi-Lo is now short. The probabilities are exactly what the buttons imply, the payout is the fair price times the configured return rounded once, and the cost of playing is one percent of everything staked — whichever button you press and however far you ride. Knowing the table does not make the game cheaper anymore. It tells you, precisely, what it costs.