Giro Games What Is a PAR Sheet? Slot Machine Maths Explained Line by Line
Every slot machine has a document that the player never sees and the regulator always asks for. In the land-based world it is called the PAR sheet — Paytable and Reel Analysis, sometimes read as Probability and Accounting Report. Online studios call it the maths spec or the model sheet, but the content is the same: a full description of how a game turns spins into money. If you can read a PAR sheet, you can read a slot.
This guide walks through a PAR sheet line by line, using the numbers of a simple 5×3, 20-line game as an example. The structure is the one we use in-house; a lab like GLI or BMM will ask for the same tables in roughly the same order.
What a PAR sheet contains
A complete slot machine PAR sheet has five parts:
- Reel strips — the exact sequence of symbols on each reel, with stop counts.
- Paytable — what each combination pays, in credits or bet multiples.
- Combination analysis — how many ways each winning combination can occur and its probability.
- Feature analysis — the same for free spins, bonuses, jackpots and any buy-feature mode.
- Summary — RTP, hit frequency, volatility index, max win probability and confidence intervals.
Line 1: reel strips and stop counts
A physical reel had 22 stops; a virtual reel can have anything from 30 to several hundred. Each strip is a list: position 1 = Ace, position 2 = King, position 3 = Wild… The count of each symbol per reel is the single most powerful lever in the maths. Put the top symbol twice on reel 1 and once on reel 5 and you have decided how often five-of-a-kind can happen before writing a single formula.
A quick worked example: reel strips of 32, 32, 34, 34 and 36 stops give 32 × 32 × 34 × 34 × 36 = 42,614,784 possible outcomes. Every probability in the sheet is a count of favourable outcomes divided by that number.
Line 2: the paytable
Pays are usually listed per line for 3, 4 and 5 symbols. A typical modern hierarchy: low symbols (card ranks) pay 0.1–0.5x bet for five, mid symbols 1–2x, high symbols 2–5x, the premium symbol 10–25x. The paytable and the strips are designed together; a beautiful paytable on the wrong strips returns 130% and a boring one on tight strips returns 80%.
Line 3: combination analysis
For each combination, the sheet lists hits (how many of the 42.6 million outcomes produce it), the probability (hits divided by total), the pay and the contribution (probability × pay). Summing contributions across all combinations gives the base-game RTP. This is the table an auditor checks first, because it can be verified by exhaustive enumeration rather than trusted on simulation.
Two details that beginners miss: wild substitution must be counted without double counting (a line that reads Wild-Wild-King is one King win, not a King win and a Wild win), and left-to-right games must exclude combinations that start on reel 2.
Line 4: feature analysis
Features are where exhaustive enumeration stops and simulation begins. The sheet states the trigger probability (for example, three scatters anywhere: 1 in 180 spins), the expected value of one feature in bet multiples, retrigger probability, and the resulting contribution to RTP. For a free-spins game a common split is 60–70% of RTP in the base game and 30–40% in the feature; games built for streamers push the feature share higher, which is exactly what makes them volatile.
If the game has a bonus buy, it gets its own column set: cost in bets, RTP of the purchased mode (often slightly higher than the base game), and its volatility. Many regulators want the buy mode documented as a separate game.
Line 5: the summary block
- RTP — total, base, feature, and per bet mode. Ours are listed to two decimals, and the same number appears on the operator’s game page. For a live example, Midas Golden Era runs at 96.05% and Sugar Space at 96.00%.
- Hit frequency — share of spins with any win. Note that “any win” includes wins smaller than the bet; a separate line for “net-positive spins” is more honest and increasingly requested.
- Volatility index — usually the standard deviation of the per-spin return. A base-game-only classic sits around 3–5; a modern feature-heavy slot sits at 8–12 and above.
- Max win — the ceiling and its probability. A 5,000x cap that hits once in 2 million spins and a 25,000x cap that hits once in 200 million are different products even if the marketing looks similar.
- Confidence interval — after N simulated spins, how far the empirical RTP may sit from the theoretical one. Labs typically expect at least 10 million spins for a video slot, and studios run billions.
How a PAR sheet is used after launch
Operators compare actual hold against the theoretical RTP; a persistent gap of more than the confidence interval means either a bug or a very unusual player mix. Regulators pull the sheet when a complaint arrives. Studios reuse it as the base for a reskin: keep the strips and paytable, change the theme, and the certification is far cheaper because the maths is already proven.
Reading someone else’s PAR sheet
Check whether feature contributions are supported by documented calculations or simulations, and whether the maximum-win probability is stated. A round-number contribution does not by itself show that the value was assumed. Nor does a hit frequency above 40% contradict high volatility: frequent small payouts can coexist with rare, very large wins. Assess the full payout distribution and the supporting evidence rather than treating either number alone as proof of a flawed model.
Building the model from the strips up — and then simulating it across billions of spins — is the part of production we do for our own games and for clients; it is described under game logic and mathematics on our services page.
Background reading (Russian): slotcatalog’s guide to RTP, variance and hit frequency, and the Chilli Heat volatility analysis on sparklix.ru, both of which describe the same metrics from the player’s side.
