Joan Ginther: four Texas Lottery jackpots, and why ‘1 in 18 septillion’ answers the wrong question

Four top prizes, one small town

On 7 August 2008 the Texas Lottery’s own milestones page recorded that Joan R. Ginther of Bishop had claimed a $3 million top prize in Millions & Millions and, with $2 million from Holiday Millionaire in 2006 and a share of an $11 million Lotto Texas jackpot in 1993, had become its first three-time top-prize winner on record. In June 2010 she won a fourth, $10 million, from the Bishop store that had sold her the third, and that July the story went national.

Bishop is a farming town of about 3,300 people, some 30 miles from the nearest claim office in Corpus Christi. AP described her as 63, a former college professor with a 1976 Stanford doctorate who had taught math at several California colleges for about a decade; by the Inquirer’s account she had moved to Las Vegas in 2001 and kept coming back to Texas to play. She asked for minimal publicity, had never spoken publicly and asked acquaintances not to give interviews. Residents told AP she had bought the church a van.

• 1993, Lotto Texas: half of an $11 million jackpot, taken as a $5.4 million annuity of 20 yearly payments of $270,000, every July through 2012.
• April 2006, Holiday Millionaire: the $2 million top prize of a $30 scratch ticket, paid as a lump sum; the Inquirer placed the sale at a Diamond Shamrock store in nearby Kingsville.
• August 2008, Millions & Millions: the $3 million top prize of a $30 ticket from Times Market in Bishop, paid as a lump sum.
• June 2010, $140,000,000 Extreme Payout: the second of the game’s three $10 million top prizes, on a $50 ticket from Times Market; about $7.5 million after taxes, officials estimated.

At face value that is $20.4 million; AP said nearly $21 million. Only the date of the 1993 draw has to be reconstructed. The Inquirer reported her numbers as 1, 4, 7, 10 and 47, which contain her birthday, 4/1/47, and the one draw in the official Lotto Texas history with all five is Wednesday 14 July 1993 (1-4-7-10-47-50), for which a third-party archive lists two jackpot tickets sharing $11 million. It is very likely her draw, but it is an inference, not a record.

Where “1 in 18 septillion” comes from

AP’s report of 13 July 2010 quoted two math professors at the University of Texas at San Antonio: the odds of four such wins were 1 in 18 septillion, an 18 followed by 24 zeros. The figure assumes that her four winning tickets were the only tickets she ever bought. On that assumption it is simply the four chances multiplied.

Lotto Texas was a 6-of-50 game until 19 July 2000: 1 in 15,890,700 a play. The game rules give 1 in 1,000,000, 1 in 900,000 and 1 in 1,200,000 for the scratch-off top prizes; ABC gave 1 in 1,028,338 for 2006. Calculation for this article: the product is 1.72 × 10²⁵ with the rules’ odds and 1.76 × 10²⁵ with ABC’s, roughly 17 to 18 septillion.

The arithmetic holds; the question is the problem. The professors made the point themselves in the same report: a habitual player winning four times over a 17-year span is much less far-fetched.

“1 in 18 septillion” answers this question: if one particular person, chosen in advance, buys exactly one ticket in each of these four games, what is the chance that all four win? Asked about Joan Ginther, it goes wrong three ways.

1. Nobody chose her in advance. The odds were computed in 2010, after the fourth win, for a player whose wins had just made the news. Judged beforehand, the surprising event is “someone in Texas collects four top prizes in 17 years”, not “Joan Ginther does”. Painting the target around the bullet holes after the shots even has a name: the Texas sharpshooter fallacy.
2. She did not buy four tickets. The estimates run to tens of thousands.
3. Scratch-off odds do not stay put. A printed game holds a fixed number of top prizes, so every ticket sold and every prize claimed changes the odds of the tickets left. The published figure is the chance at the start.

The Inquirer, which rebuilt her record in 2013 and 2014, made the same point: AP’s figure gives the odds for this particular person buying exactly four tickets, not the odds that such a thing would ever happen. The University of Houston’s math department chair told KSAT in 2012 that for one person the chance was, in KSAT’s paraphrase, basically zero, but that spread over 20 million people like her and 20 years it is not so crazy.

The question that matters: how many tickets

From April 2013 the Inquirer filed more than 60 right-to-know requests. Records from the Texas Lottery and the state Comptroller turned up 25 more claims of $1,000 to $3,000 between 2005 and 2012: 28 instant prizes of $1,000 or more in at least eight games, with none after 2012, the year of the last $270,000 annuity payment. There is no count of her tickets, only estimates:

• More than 80,000 tickets, more than $2.5 million: the average scenario of two statistics experts the Inquirer consulted; under other assumptions its estimates ran from 72,000 tickets ($2.4 million) to about 100,000 ($3.3 million).
• About 3,000 tickets a year: what people in Bishop estimated, as reported by Harper’s in 2011.
• Whole packs, even rolls, of $50 tickets: what store clerks described.

The store where she bought her last two winners points the same way. Times Market sold 9,380 Extreme Payout tickets, $469,000 worth, by far the most of any store in Texas; no other store sold 6,000. In one week of October 2009 it received a fifth of all the Extreme Payout tickets the Lottery shipped that week.

Put that volume into the question. If each ticket is an independent chance at a top prize, the number of top prizes from N tickets is close to a Poisson count with mean N times that chance.

Poisson approximation, the same independent chance on every ticket. 1 in 1M is close to the printed odds of the three games she won; 1 in 600K is what the Inquirer said her odds often were, 1 in 200K what they perhaps reached. Her real tickets spanned at least eight games with different odds, so the rows show the scale, not her history.

An industry analyst quoted by the Inquirer, assuming 104,000 tickets at average odds at least double the published ones, got better than 1 in 10 for one top prize and about 1 in 1,300 for three. Either way the scale has moved from septillions to thousands, and at thousands “someone” is the right word: if 2,800 players each bought like the middle row, the chance that at least one of them collects three top prizes is about 63% (calculation for this article). How many Texans bought that heavily is not on the record, and how surprising her run was depends on exactly that number.

Volume cuts both ways. In Ultimate Casino Jackpot, a $50 game launched in 2010, her pack numbers came within about 1% of the $7.5 million winner, but the tickets in between would have cost more than $125,000 even after her tax break. And the Inquirer judged that she was probably a major loser in 2011 and 2012.

What a $50 ticket was worth

A scratch game’s rules tell you what it pays back. The prizes are fixed before a ticket is sold, so the return is every prize added up and divided by the number of tickets times the price.

Return = all prizes ÷ (tickets × price): $65,250,000 over about 3.0 million $30 tickets, $78,345,000 over about 3.6 million, and for Extreme Payout the closing analysis’s $143,554,845 over 3,707,320 $50 tickets (77.44% printed, 78.18% actually paid; at launch Texas called its 78% the highest ever for a Texas Lottery instant game). Tops gone = the return once the top prizes have been claimed.

A $50 Extreme Payout ticket returned $38.72 on average, an expected cost of $11.28 before tax. About $8 of that was the three $10 million prizes; the other $30.63 was everything else, and it came out evenly: by December 2010, with 60.68% of the tickets sold, each prize tier below $10,000 had paid out 58.75% to 61.22% of its winners. The small prizes arrive in step with sales, whoever buys. The top prizes do not. Until each is sold it sits somewhere among the unsold tickets, and that is what makes a scratch game a pool rather than a draw.

Worth = $30.63 of smaller prizes per ticket plus $10 million for each top prize still out, spread over the unsold tickets, before tax, assuming a remaining top prize is equally likely to be in any of them. The 39% row is the game’s state in December 2010: 60.68% sold, one top prize left.

On those figures a remaining ticket was worth its full $50, before tax, only when top prizes were still out late in the game: all three with no more than about 42% of the tickets unsold, two with 28%, one with 14%. Count tax on the top prize alone, at the officials’ estimate that her $10 million came to about $7.5 million, and the thresholds fall to about 31%, 21% and 10% (calculations for this article). That is why the count of top prizes left is worth knowing. Texas began publishing it in January 2006, a few months before her first scratch-off top prize, though a players’ newsletter had printed the same counts for years; the pack numbers of top-prize winners were not shared until the fall of 2012. The Inquirer found that she rarely bought from the first 30% of a print run and often played when her odds had improved to about 1 in 600,000, perhaps even 1 in 200,000.

The rules also let a game keep selling after every top prize is gone. When staff recommended closing Extreme Payout in December 2010, one $10 million prize was still outstanding and 1,337,780 tickets were still in the warehouse. No source found for this article says whether that last top prize was ever claimed.

What the tickets cost her was never published. The Inquirer offered an illustration under the federal rules of 2005 to 2012, a 35% top rate and gambling losses deductible against gambling winnings: at a 60% average return, $375,000 of tickets a year brings back $225,000, and the $150,000 shortfall, written off against other gambling winnings, costs about $97,500 after tax, roughly 25 cents on the dollar. The write-off needs other gambling winnings to set against, and until 2012 she had a $270,000 annuity every July. It is a model of what the volume could have cost, not a record of what it did.

What is documented, and what was only suggested

In “The Luckiest Woman on Earth” (Harper’s Magazine, August 2011), Nathaniel Rich speculated that she might have worked out how each game’s major prizes were seeded through its print run and, with the shipping schedule to stores apparently fixed, predicted where winners would land. He also reported residents’ complaints that she monopolized some games. Rich offered it as a theory, and it was never proven. That year the Texas Lottery told the Houston Press that no allegations or hints of wrongdoing had been brought to its attention and that no investigation had been conducted.

On the record

• Four top prizes in her name, 1993 to 2010 (the Texas Lottery’s milestones page, through 2008; AP; ABC).
• 28 instant prizes of $1,000 or more, 2005 to 2012, none after 2012 (records obtained by the Inquirer).
• Extreme Payout’s print run, prize fund, sales and outstanding top prize (the Lottery’s closing analysis).
• No investigation (the Lottery to AP in 2010 and to the Houston Press in 2011).

Estimated

• Her volume: more than 80,000 tickets and $2.5 million in the experts’ average scenario; 72,000 to about 100,000 tickets under other assumptions.
• Her odds per ticket: often about 1 in 600,000.
• Her 1993 draw: very likely 14 July 1993.

Suggested, never shown

• That she had worked out how prizes were placed and where they would ship (Rich, Harper’s, 2011).
• That her father bought the 1993 ticket: a rumor that KSAT said could not be independently confirmed.

A scratch-ticket maker doubted the theory. Asked how many lottery workers know where the unscratched winners are, the vice-president of game development at Scientific Games told the Inquirer: “Nobody. Zero.” No amount of computing power, he said, would predict the next ticket. Mohan Srivastava, who had found that in some scratch games the symbols printed on the outside of a ticket gave clues to the winners, said his method could not have worked in Texas at the time and that her tickets showed no telltale patterns, though he remained skeptical that she was simply lucky; an organizer of the MIT group that played Cash WinFall struggled to find a plausible explanation. The Inquirer’s conclusion was plainer: she won chiefly because she bought an extraordinary number of tickets.

The Lottery itself never looked further. It told AP in 2010 that it had never investigated her winnings for possible fraud, an official said she had obviously been born under a lucky star, and by 2014, the Inquirer reported, it had never even asked her how she won. She was never charged with anything. She died in April 2024, at 77, according to an obituary, never having answered the Inquirer’s requests.

A scratch card with nothing left to count

Most of what is unusual in this story comes from one property of printed scratch games: each is a finite pool whose odds move with every ticket sold and every prize claimed. The remaining-prize counts, the pack numbers, the warehouse and the theories about shipping schedules all exist because the pool can be counted down.

Betkyo’s omikuji is a scratch card without a pool. There is no print run, no pack and no count of prizes left. Each card is one independent draw from the server seed, whose SHA-256 hash is shown before you bet, your own client seed and the card’s nonce; the tier is the first number off that stream, scanned against fixed weights, before any symbol is painted. Nothing is used up, so the thousandth card has exactly the chances of the first.

Two things follow. Volume buys more independent chances at the same odds, and timing buys nothing, because there is no late moment and no count worth keeping. And more cards buy more of the same 96.55%: the 3.45% edge does not shrink with volume, it shows up more reliably, as how many bets before the edge shows works out for dice. Set 96.55% beside a $50 ticket that returned 77 cents on the dollar and you have a comparison of prices, not a reason to play.

Her four wins were never 1 in 18 septillion. The three scratch-off prizes came to a very heavy buyer playing pools that could be counted down, with a large helping of luck on top: about 1 in 1,300 for three top prizes on the Inquirer analyst’s assumptions, about 1 in 2,800 on this article’s middle row. The right question about any improbable win is not how unlikely it was for the winner. It is how many people were trying.