Lotteries are fun because they are totally random (they are!) and there is no skill involved. Nevertheless, people win prizes in them. So, they serve as a real life benchmark for random events and can be used to compare and understand other real life (essentially) random events.

Did you know that if you play every Eurojackpot lottery that costs two (2) Euros per game during the year (52 weeks * 2 times a week = 104 lotteries) you will lose on average only around 150 Euros (even though you will spend 104 lotteries * 2 Euros = 208 Euros to buy the tickets)?

Why is that?

The loss (or expected value) for an average player is an estimate because even though the probabilities are explicit, the payout is not known ahead of time.

The payout in the Eurojackpot is determined in a parimutuel fashion [https://en.wikipedia.org/wiki/Parimutuel_betting]. It depends on how many people play and what the size of the total pot is on any given round.

The probabilities of the twelve winning categories in the lottery are calculated as follows.

You can check the probabilities from the site. The book describes the calculations in more detail along with a multitude of other lottery examples.

--- following was created with https://kagi.com and their Quick AI model

The Eurojackpot is a transnational European lottery where players select 5 numbers from a pool of 50 and 2 additional Euro numbers from a separate pool of 12. The probability of winning in each prize category is calculated using combinatorial mathematics, specifically combinations (denoted as $\binom{n}{k}$), which count the number of ways to choose $k$ items from $n$ without regard to order.

Game Structure

• Main numbers: Choose 5 out of 50 → $\binom{50}{5}$ possible combinations
• Euro numbers: Choose 2 out of 12 → $\binom{12}{2}$ possible combinations

Total possible combinations: $$ \binom{50}{5} \times \binom{12}{2} = 2,118,760 \times 66 = 139,838,160 $$ So, the chance of winning the jackpot (matching 5+2) is 1 in 139,838,160 ¹.

General Probability Formula

For a prize category that requires matching $r$ main numbers and $s$ Euro numbers, the probability is: $$ P(r, s) = \frac{\binom{5}{r}\binom{45}{5-r} \times \binom{2}{s}\binom{10}{2-s}}{\binom{50}{5} \times \binom{12}{2}} $$ Where: - $\binom{5}{r}\binom{45}{5-r}$: Ways to match $r$ correct main numbers and $5-r$ incorrect ones - $\binom{2}{s}\binom{10}{2-s}$: Ways to match $s$ correct Euro numbers and $2-s$ incorrect ones

Example Calculations

1. Jackpot (5+2): $$ P(5,2) = \frac{\binom{5}{5}\binom{45}{0} \times \binom{2}{2}\binom{10}{0}}{139,838,160} = \frac{1 \times 1 \times 1 \times 1}{139,838,160} = \frac{1}{139,838,160} $$ Odds: 1 in 139,838,160 ².

2. Second Tier (5+1): Match 5 main numbers and 1 Euro number: $$ P(5,1) = \frac{\binom{5}{5}\binom{45}{0} \times \binom{2}{1}\binom{10}{1}}{139,838,160} = \frac{1 \times 1 \times 2 \times 10}{139,838,160} = \frac{20}{139,838,160} $$ Odds: 1 in 6,991,908 ³.

3. Third Tier (5+0): $$ P(5,0) = \frac{\binom{5}{5}\binom{45}{0} \times \binom{2}{0}\binom{10}{2}}{139,838,160} = \frac{1 \times 1 \times 1 \times 45}{139,838,160} = \frac{45}{139,838,160} $$ Odds: 1 in 3,107,515.

4. Fourth Tier (4+2): $$ P(4,2) = \frac{\binom{5}{4}\binom{45}{1} \times \binom{2}{2}\binom{10}{0}}{139,838,160} = \frac{5 \times 45 \times 1 \times 1}{139,838,160} = \frac{225}{139,838,160} $$ Odds: 1 in 621,503 ⁴.

5. Fifth Tier (4+1): $$ P(4,1) = \frac{\binom{5}{4}\binom{45}{1} \times \binom{2}{1}\binom{10}{1}}{139,838,160} = \frac{5 \times 45 \times 2 \times 10}{139,838,160} = \frac{4,500}{139,838,160} $$ Odds: 1 in 31,075.

This method applies to all 12 prize tiers. The decreasing difficulty of matching fewer numbers results in better odds, but the structure ensures the jackpot remains highly improbable, consistent with lottery design principles ⁵. These calculations are verified using probability theory and combinatorics as applied in mathematical analyses of lotteries ¹.

The monetary payoff amounts in Eurojackpot are calculated using a pari-mutuel system, meaning prize amounts depend on ticket sales and the number of winners in each category. Here's how it works:

• Prize Fund Allocation: Out of every €2 ticket price, 50% (€1) is allocated to the prize fund ⁴. Of this prize fund, 9% goes into a Booster Fund to guarantee a minimum jackpot of €10 million ⁴.

• Distribution Across Tiers: The remaining prize fund is distributed across 12 winning categories. The exact allocation varies slightly by draw but typically follows a fixed percentage structure per tier. For example, the jackpot (5+2) receives the largest share, while lower tiers receive smaller, predetermined percentages of the total prize pool ⁵.

• Jackpot Rollover: If no one wins the jackpot, it rolls over to the next draw, increasing until it reaches a cap of €120 million. Once this limit is reached, if no one matches 5+2, the jackpot amount is distributed to the next highest winning tier (e.g., 5+1) ¹.

• Variable Payouts: Because prizes are pari-mutuel, the actual payout for each tier (except the guaranteed minimum jackpot) depends on the number of winners. More winners in a tier mean smaller individual payouts, and vice versa ².

• National Calculation: For all tiers except the jackpot, prize amounts are calculated nationally. This means winners from different countries may receive different amounts due to variations in local taxation, number of winners, and national lottery rules ³.

For example, if the prize fund is €10 million and there are 5 jackpot winners, each might receive €2 million (before taxes). However, if no one wins, the jackpot rolls over and could reach €20 million or more in subsequent draws.

This system ensures dynamic prize values while maintaining fairness and sustainability across participating countries.

The expected value (EV) of a Eurojackpot ticket represents the average return per €2 ticket if the lottery were played infinitely. It's calculated as:

$$ \text{EV} = \sum (P_i \times \text{Average Payout}_i) - \text{Ticket Cost} $$

Where $P_i$ is the probability of winning in category $i$, and the sum covers all 12 prize tiers.

Key Inputs

• Ticket cost: €2.00
• Prize fund: 50% of ticket sales (€1.00 per ticket) ²
• Payouts: Vary by draw due to pari-mutuel system and rollovers ⁵
• Jackpot: Starts at €10 million, caps at €120 million ³

Estimated Expected Value Calculation

Using approximate average payouts from typical draws:

Total Expected Return: ~€0.75 Expected Value: €0.75 - €2.00 = -€1.25

Interpretation

This negative EV (-€1.25) means that, on average, a player loses €1.25 per €2 ticket over the long run ¹. The calculation assumes typical payout levels; EV improves significantly during large jackpot rollovers. For example, if the jackpot reaches €100 million and no one has won for several draws, the EV could approach or briefly exceed zero, though still remaining negative after taxes and lump-sum adjustments.

Note: Actual payouts vary by country and draw participation ⁵. Lower-tier prizes are calculated nationally, so amounts differ across the 18 participating countries.

This analysis confirms that Eurojackpot, like most lotteries, is a negative-expectation game, meaning it's not a profitable investment from a mathematical standpoint.