One card, 38 balls
The game is 75-ball bingo played alone. The engine deals a 5×5 card — five numbers from 1–15 in the B column, 16–30 in I, and so on — marks the centre cell free, then draws 38 of the 75 balls. Every row, every column and both diagonals is a line, twelve in all, and the payout depends only on how many of the twelve are complete. Designing Bingo explained why 38 balls and why those seven pay steps; it quoted the line-count distribution from a million simulated rounds. This article computes it exactly.
One fact makes the exact calculation tractable. The card holds 24 real numbers, and which of them get marked depends only on which 38 balls are drawn: a specific set of k card numbers being marked, and the other 24 − k not, has probability C(51, 38 − k) ÷ C(75, 38), the same for every card layout. So the distribution of line counts is the sum over all 2²⁴ = 16,777,216 mark patterns of that weight — a few seconds of computation, and no sampling error.
Every line count, priced
| LINES | CHANCE | ONE IN | PAYS | SHARE OF RETURN |
|---|---|---|---|---|
| 0 | 62.82% | 1.6 | — | — |
| 1 | 28.60% | 3.5 | ×1.5 | 42.89% |
| 2 | 7.059% | 14.2 | ×4 | 28.24% |
| 3 | 1.288% | 77.6 | ×12 | 15.46% |
| 4 | 0.2007% | 498 | ×30 | 6.02% |
| 5 | 0.02913% | 3,433 | ×100 | 2.91% |
| 6 | 0.004060% | 24,600 | ×200 | 0.81% |
| 7 or more | 0.000645% | 155,000 | ×500 | 0.32% |
| Total | 100% | 96.66% return · 37.18% hit rate |
Share of return = chance × payout, as a percentage of the stake. The design notes’ simulated figures (62.9 / 28.6 / 7.05 / 1.27 / 0.20 / 0.03%, return ≈ 96.5%, hit 37.1%) sit within sampling error of these; the exact house edge is 3.34%.
Two things stand out in the table. First, the money is at the bottom of the paytable: the single-line ×1.5, hit on two cards in seven, carries 43% of the return, and one and two lines together carry 71%. The ×100 and above rungs that give the game its top are worth 4% of the return between them. Second, the steps are steep on purpose: each extra line is roughly five to seven times rarer than the last, and the payout roughly triples, so the return per rung falls with rarity rather than staying flat. That is a design choice — Keno makes the same one — and it is what keeps the whole card at 96.66% while the rare counts stay memorable.
| LINES | ONE IN |
|---|---|
| 7 | 179,000 |
| 8 | 1.30 million |
| 9 | 12.1 million |
| 10 | 67.5 million |
| 11 | impossible |
| 12 (every line) | 2.67 billion |
Eleven is impossible because every cell sits on one row and one column, so a single unmarked number always breaks at least two lines: line counts are lost two at a time, never one. Ten lines is one unmarked number off both diagonals; twelve means all 24 numbers drawn, C(51, 14) ÷ C(75, 38).
The free star and the average card
A ball lands on any given card number with probability 38 ÷ 75 = 50.67%, but a line needs all of its numbers, and the draws are without replacement. A line that avoids the centre needs five specific balls among the 38: 38 ÷ 75 × 37 ÷ 74 × 36 ÷ 73 × 35 ÷ 72 × 34 ÷ 71 = 2.908%, one in 34.4. A line through the free star needs only four: 6.073%, one in 16.5. Four of the twelve lines pass through the star — the middle row, the middle column and both diagonals — so the expected number of completed lines on a card is 8 × 2.908% + 4 × 6.073% = 0.476.
| MARKED | CHANCE |
|---|---|
| 8 | 2.44% |
| 10 | 11.20% |
| 12 | 19.46% |
| 13 | 17.96% |
| 14 | 13.07% |
| 16 | 3.33% |
| 18 | 0.30% |
| 20 or more | 0.0095% |
Hypergeometric: C(24, k) × C(51, 38 − k) ÷ C(75, 38). The mean is 12.16 marks; the card looks half full on an average round, which is why one line is common and three is not.
The marked-number table is the honest picture of a typical round: about twelve of the twenty-four numbers get a mark, scattered across the grid, and the question is only whether five of them happen to sit in a row. Fourteen or more marks happen one round in four and still usually complete at most a line or two; the three-line card, one in 78, is nearly always a card with an unusually lucky arrangement rather than an unusually full one.
Because the line counts and the payouts are fixed, the per-round swing is fixed too: the standard deviation of one round’s result is 3.26 stakes. Over 100 one-unit rounds the expected loss is 3.3 units and the swing about 33 units, so a session that ends 30 units up or down is ordinary. Risk of ruin turns that into how long a given bankroll usually lasts.
Checking a card
- From the round’s fairness panel take the revealed server seed, the client seed and the nonce.
- For cursors 0 to 24 compute HMAC-SHA256 keyed by the server seed over “clientSeed-nonce-cursor”, take the first 8 bytes as a fraction of 2⁶⁴, and walk the card column by column: each draw picks index floor(u × remaining) from that column’s fifteen numbers, swapping the last number into the hole. Cell 12 is then the free star.
- For cursors 25 to 62 do the same pool hop over 1–75 to get the 38 balls, mark the card, count the complete lines among the twelve, and read the payout from the table above.
The open-source verifier’s bingo command does the three steps in one call. The layout and the balls come from one committed seed pair, so a card cannot be dealt after the balls are known, and the balls cannot be drawn to fit the card — which is what the exact distribution above assumes, and what the commitment lets you check.
What are the odds of getting a bingo line?
On this 38-of-75 solo card, 37.18% of rounds complete at least one line: 28.60% exactly one, 7.06% two, 1.29% three, 0.20% four, 0.029% five, 0.0041% six, and seven or more once in 155,000.
What is the house edge in solo bingo?
Exactly 3.34%: the return is 96.66% of the stake, computed over every possible mark pattern rather than simulated. The design notes’ simulated 96.5% was an estimate of this number.
Why is the free centre worth a whole ball?
A line through the centre needs four drawn numbers instead of five, so it completes 6.07% of the time against 2.91% — about twice as often. Four of the twelve lines pass through it.
Can you complete 11 lines?
No. Every cell is on one row and one column, so any unmarked number breaks at least two lines at once; you can lose lines two at a time but never exactly one, and the count jumps from ten (one unmarked number off the diagonals) to twelve. All twelve happens once in 2.67 billion cards.
How many numbers on the card usually get marked?
On average 12.16 of the 24. Twelve marks exactly is the single most likely outcome (19.5%); twenty or more happens once in about 10,500 rounds.
Which payout gives most of the return?
The single line at ×1.5, which pays on 28.6% of rounds and accounts for 42.9% of the 96.66% return. One and two lines together are 71% of it; everything from ×100 up is 4%.
Can I verify a bingo round?
Yes. One HMAC-SHA256 per cursor keyed by the revealed server seed over “clientSeed-nonce-cursor”: cursors 0–24 deal the card, 25–62 draw the balls, and the verifier’s bingo command reproduces both and the line count.
- Betkyo engine source, house service: BingoEngine.kt — DRAWS, FREE_CELL, LINE_PAY100, LINES, play() (card by column-major pool hop on cursors 0–24, balls on cursors 25–62, pay100 by completed lines); OriginalBingoService.kt settlement
- Betkyo engine source, client mirror: playBingoRound(), BINGO_LINE_PAY100 and BINGO_LINES in bingo/derive.ts
- Exact calculation written for this article: the 2²⁴ mark patterns of the 24 card numbers, each weighted by C(51, 38 − k) ÷ C(75, 38), summed by completed-line count; the single-line probabilities 38·37·36·35·34 ÷ (75·74·73·72·71) and 38·37·36·35 ÷ (75·74·73·72); the hypergeometric marked-number distribution; per-round standard deviation
- Betkyo provably-fair verifier (GitHub, MIT): the bingo command reproduces the card, the balls and the line count from serverSeed, clientSeed and nonce
