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Rollover ticket expected value: jackpot, sales, tax and shared wins
When nobody wins a lottery's top prize for a few draws, the prize rolls over and grows. The news reports "this week's jackpot is ○○", and people ask: "at this size, isn't it worth buying?" This page answers that question with calculation. It does not fix a conclusion in advance; it states the numbers the model gives. What the model assumes and what it leaves out matter as much as the results, so assumptions and limits are set out at length.
The model: a ticket's expected value
A ticket's expected value is the sum of its expected values for each prize. Lower prizes are nearly fixed or have many winners and are stable; the effect of a rollover is concentrated on the jackpot. The jackpot part is calculated like this.
- The chance of the jackpot is 1/N. N = 8,145,060 for Lotto 6/45 and N = 292,201,338 for Powerball.
- If S tickets are sold in a round, the expected number of jackpot winners is λ = S/N.
- If I win the jackpot, take the number K of other winners as Poisson with mean λ (assuming everyone picks numbers uniformly at random).
- My share is 1/(1+K) of the jackpot, and its expected value is (1 − e−λ) ÷ λ.
- If there is tax, average the after-tax share (with progressive tax this differs from applying tax to the average).
In this model the jackpot expected value per ticket is (1/N) × (average after-tax share × jackpot). It is the same formula seen in the split-risk article. When sales rise, λ rises and the share falls, partly offsetting the effect of the larger rolled-over prize.
Calculator
Choose a lottery model and enter the jackpot (cash value for Powerball, rollover for Korea), tickets sold and an assumed tax rate. The defaults are illustrative.
Powerball: results
Using the amounts and odds on the official prize chart, the expected value of all non-jackpot prizes on one $2 Powerball ticket is about $0.32 (the sum of the eight tiers from five matches, $1 million at 1 in 11,688,053.52, down to the Powerball alone, $4 at 1 in 38.32). The rest is the jackpot part. Assuming 150 million tickets sold and varying the jackpot cash value gives:
| Jackpot cash value | Expected value (before tax) | Expected value (40% tax assumed on the jackpot) |
|---|---|---|
| $100 million | $0.59 | $0.48 |
| $200 million | $0.86 | $0.64 |
| $400 million | $1.39 | $0.96 |
| $600 million | $1.93 | $1.28 |
| $834.9 million (Dec 2025 case) | $2.55 | $1.66 |
| $1.2 billion | $3.53 | $2.25 |
The point at which expected value passes the $2 price (the break-even cash jackpot) depends on sales.
| Sales assumed | Before tax | With 40% tax assumed on the jackpot |
|---|---|---|
| 50 million tickets | about $534 million | about $890 million |
| 150 million tickets | about $628 million | about $1,046 million |
| 300 million tickets | about $785 million | about $1,309 million |
What the calculation shows:
- Before tax, in a very large rollover round with a cash value above roughly $500–800 million, the model's expected value of a ticket exceeds its price. At a cash value of $834.9 million and 150 million tickets sold it is $2.55, 1.28 times the price.
- With an assumed tax (40% on the jackpot) the same round falls to $1.66, below the price, and break-even rises to about $0.9–1.3 billion. The 40% is an illustrative assumption, not a verified US federal and state rate. Real rates differ by state and by winner.
- Higher sales lower the expected value for the same prize, because the prize is divided more.
The conclusion cannot be put in one line. In a very large rollover, expected value can pass the price before tax, and the conclusion flips depending on tax and sales assumptions. This is not advice that "at this prize one should buy". Even when expected value exceeds the price, the chance of winning remains about 1 in 292 million, and any single ticket's result almost always differs from expectation: expected value is an average over very many tickets.
Korean Lotto 6/45: results
In Korea the prize pool is fixed at 50% of sales and a rolled-over first prize is added to the next round's first prize. The model takes 50% of sales, subtracts the fixed fourth and fifth prizes (about 18% in expected value), splits the rest 75/12.5/12.5% among the first, second and third prizes, and adds the rollover to the first. Tax uses this site's tax-guide rules (KRW 2 million minimum, 20% up to KRW 300 million and 30% above, 10% local income tax) on the first to third prizes. It is calculated at about the sales of round 1247 (roughly 118.98 million games), varying the rollover.
| Rollover added to the first prize | Expected value | Of the price |
|---|---|---|
| KRW 0 (no rollover) | KRW 416 | 41.6% |
| KRW 10 billion | KRW 473 | 47.2% |
| KRW 20 billion | KRW 529 | 52.9% |
| KRW 40.8 billion | KRW 646 | 64.6% |
| KRW 100 billion | KRW 979 | 97.9% |
| about KRW 103.7 billion | about KRW 1,000 | 100% |
After tax and with no rollover, a ticket's expected value is about KRW 416 (before tax it is KRW 500, so tax takes about KRW 84). The rollover must reach about KRW 103.7 billion for expected value to reach the KRW 1,000 price. Compare that with the data: in rounds 88 to 1247 the largest first-prize total (amount per winner × winners) was about KRW 43.6 billion in round 464 (after a previous round with no winner), while the model's break-even needs a first-prize total of about KRW 132.2 billion (about KRW 28.5 billion for the base pool plus KRW 103.7 billion of rollover), three times that. The median was about KRW 15.4 billion and the 95th percentile about KRW 28.7 billion.
So within this model and this record, Korean Lotto's expected value never passed its price even with a rollover. Even the largest round, KRW 43.6 billion, comes to about KRW 600–650 in the model (KRW 598 with 150 million games sold). This depends on the fact in this data that rollovers in Korea did not build up to hundreds of millions of dollars as Powerball's can; if rules change or a bigger rollover arises, the conclusion could change.
Assumptions and limits (before reading the results)
- Uniform number choice. The model assumes everyone picks numbers uniformly at random. In this site's step-1 analysis real first-prize counts scattered 1.73 times more than Poisson expects (winner-count analysis). The effect of overdispersion goes two ways. For an average combination the expected share is slightly larger than the Poisson model gives (when the number of co-winners scatters more, the mean of 1/(1+K) rises; this is the mathematical direction only and we did not compute its size). If instead you pick a combination many people choose, the share is smaller (the birthday-number trap). This calculator reflects neither.
- Sales are an assumption. Sales rise with the jackpot, but we did not confirm that relationship. Enter sales yourself to see the sensitivity (the three sales figures in the Powerball table).
- Cash value was used. For Powerball the cash value is taken as the jackpot. If you choose the annuity, a present-value calculation is needed (lump sum versus annuity).
- Tax. Powerball's 40% is an illustrative assumption. We did not check US tax law. For Korea the tax-guide rules were applied, with the ticket cost expense set to zero and truncation to the won assumed. Tax on lower prizes is not applied for Powerball.
- Power Play and other options are not included.
- Korean prize structure. The model follows the operator's description that fourth and fifth prizes are fixed and the first to third share the rest in proportion, and that the prize pool is 50% of sales. The actual payout of fourth and fifth prizes wobbles with winner counts and was replaced by its expectation. Legal limits and rules of rollover (such as for how many rounds a prize may roll over) are not modelled.
- Average versus one ticket. Expected value is an average over very many tickets. One ticket's result is almost always zero or a small amount.
How to read the results
- An expected value below the price means a loss on average, and one above the price does not mean "you can make money". With a jackpot chance this small, most buyers lose whatever the expected value.
- A rollover does not change the probability. It changes the size of the first prize and the number of people buying. The two act in opposite directions.
- This calculator is not a tool to encourage or discourage buying. For responsible play see here.
This page is general education, not tax or legal advice. Tax rules change; check the cited statute or a licensed tax professional before acting.
Sources and verification
- Powerball prize chart: prize amounts and odds of the eight lower tiers, jackpot odds 1 in 292,201,338, overall odds 1 in 24.87 for a $2 play — verified · the page lacked the match column, so rows were matched to the standard combinations by order
- Donghaeng Lottery Lotto 6/45: prize pool 50% of sales, 4th and 5th fixed, 1st/2nd/3rd get 75/12.5/12.5% of the rest — verified
- LottoFlick aggregate: first-prize counts scatter 1.73 times more than Poisson — computed by LottoFlick
- LottoFlick aggregate: payout over sales 50.0%; first-prize pool per round — computed by LottoFlick
- Korean lottery prize tax guide (2.0 million minimum, 20%/30% bands, 10% local tax) — verified
- Powerball news, 25 Dec 2025: jackpot $1.817 billion, cash option $834.9 million (used as an example jackpot) — verified