Lottery
Secondary Prizes
The jackpot is a lottery; the small print is a price list. Set the three secondary prizes, choose how they are funded, and drag the ticket count to see whether the expected-value line merely shifts — or actually bends.
The shells
Seven positions, each drawn with replacement from 1–19, give N = 197 = 893,871,739 tickets. A ticket matches m positions when m slots carry the winning number and the other 7 − m slots carry one of the 18 wrong ones, so the tickets at each tier number C(7, m) × 187−m:
| Tier | Tickets on shell | Odds per ticket | Fair prize |
|---|---|---|---|
| Match 7 | 1 | 1 in 893,871,739 | N |
| Match 6 | 126 | 1 in 7,094,220 | $7,094,220 |
| Match 5 | 6,804 | 1 in 131,374 | $131,374 |
| Match 4 | 204,120 | 1 in 4,379 | $4,379 |
The fair prize for a tier is N divided by its shell — the amount at which that tier, on its own, would exactly pay for the whole ticket space. A tier paying x% of fair returns x cents on the dollar.
Pricing the whole table
Buy every ticket and you win every prize exactly once. The secondary table is worth
and the expected profit on k distinct $1 tickets is
The small print adds a flat Tsec / N per ticket — 15.0 cents at the defaults — whatever k is. It shifts the line. It does not bend it.
Three ways to write the small print
Fixed prizes. Every qualifying ticket is paid in full. Per-ticket secondary return r = Tsec / N, independent of k. A pari-mutuel pool funded from the same draw’s sales behaves the same way in expectation, because the pool grows with your purchases.
Capped pool. Each tier’s pool is sized to pay the public’s expected winners at the listed prize — a liability cap. Your k tickets then share it pro rata with the public’s T:
Roll-down. Prizes are fixed, and if nobody hits the jackpot the net jackpot is split among the secondary winners — a pool funded by earlier draws, dropped into this one. With the public holding T random tickets, the chance nobody wins is P0 = (1 − k/N)(1 − 1/N)T, and your expected share of the roll-down is k / (k + T):
Only a pool that is not funded by the tickets you are buying can lift a ticket above its price. That is the whole history of Cash WinFall in one line.
The default numbers
- Advertised jackpot: $1,000M → 60% lump sum, 37% tax → $378.0M net
- Secondary table: $500,000 / $7,500 / $100 → Tsec = $134.4M → 15.0¢ per dollar
- Tickets purchased: 446,935,870 (the 50% purchase from the earlier puzzle)
- EV per ticket: 42.3¢ + 15.0¢ − $1 = −42.7¢
- Break-even advertised jackpot: $2,009M (was $2,365M without the small print)
- Multiplier to break even at $1,000M: 3.84× on every secondary prize
Things to try
- In Fixed mode, drag Tickets from 1M to 893.9M. The curve is a horizontal line — that is the bonus question.
- Push Match 6 to $1,925,000, Match 5 to $28,750 and Match 4 to $385 (each about 3.85× its default). The verdict turns green at every purchase size.
- Switch to Capped pool and drag Tickets up. Your per-ticket return slides toward zero as you crowd out the public.
- Switch to Roll-down with 100M public tickets and drag Tickets down to 10M. Then set the public to 0 and watch the half-lottery purchase land a hair under break-even.
Secondary Prize Calculator
Set the jackpot, the prize table, and the rule for funding it. Then change how many tickets you buy.