Lottery Prize Formula - Scratch Card

Question

My math got rusty :( I'm looking to support our local charity with scratch cards. I need to create a formula to calculate the prize breakdown automatically. I try to explain it, I hope it's possible/clear, but my brain is fried:

• "Winning odds" need to be 50%. So 50% of scratch cards must win a prize.
• The "prize pool" is 50% of the "proceeds".
• I have a total of 5 "prize tiers".
• The "1st tier" will be 10% of the "prize pool", capped at 10 000 USD.
• The "1st tier" has only one winning scratch card.
• Tiers 2-5 have multiple winning scratch cards.
• Now I need to redistribute the remaining "prize pool" between the remaining 4 "tiers", whilst adhering to 50% odds of winning.

Example:

Proceeds: 100 000 scratch cards sold for 2 USD each = 200 000 USD
Prize pool: 50% of "proceeds" = 100 000 USD
Odds: at 50% = 50 000 scratch cards must win a prize
1st tier: at 10% of "prize pool" capped at 10 000 USD = 10 000 USD on 1 winning scratch card

Things are clear up to this point and I can't figure out how to do the rest:

2nd tier: each scratch card wins 10 USD
3rd tier: each scratch card wins 5 USD
4th tier: each scratch card wins 2 USD
5th tier: each scratch card wins 1 USD

Now I need to redistribute the remaining 90 000 USD between prize tiers 2-5 on an additional 49 999 winning scratch cards. This needs to work dynamically for any amount of proceeds, not just the above example.

Anyone who can help?

Answer 1

You have here 8 variables (percentage of winners of each tier, and the amount they win) with only two constraints: the total percentage of winners and the prize pool. You need to add some more constraints or assumptions, otherwise, there are just too many options. Follows my example:

Let $x$ be the percentage of winners in tier 2. I want them to be rare, so there will be only $2x$ winners in tier 3, $4x$ in tier 4, and $8x$ in tier 5 (don't like it? choose your own ratios that seem appropriate to you). This means that the total percentage of winners is $x+2x+4x+8x=15x=50%$ (I neglect the sole winner in tier 1). Hence, $x=3.33\%$ and we have the winning probabilities in each tier.

Now, for the prize. Using the same logic, let $y$ be the winning prize in tier 5. In tier 4 I want people to get $2y$, in tier 3 get $5y$ and in tier 2 $10y$. Once again, any numbers here will do. So the total given prizes here are $y\cdot 8x+2y\cdot 4x + 5y\cdot 2x + 10y \cdot x$ which is almost $120y$. The total prize fund is $A$, which is either 40% if the top tier got 10% or the appropriate percentage if he got the cap of 10K. Thus, $y=\tfrac{A}{120}$.

You can now compute the results in your example, see if they fit the logic and maybe adjust the ratios so it will seem nicer for you and for the players.

Answer 2

Here is one simple method:

2x + 5y + 10z = 90000 - 1w

and

w + x + y + z = 49999 .

1w = 22500 as 22500 tickets; 2x = 22500 as 11250 tickets; 5y = 22500 as 4500 tickets; 10z = 22500 as 2250 tickets.

But the total number of tickets here is only 40500. So w is increased in steps of the percentage that the number of tickets are low. Then x, y, and z are changed only relative to the revised pot for x, y, and z.

1w = 35452 as 35452 tickets; 2x = 18183 as 9092 tickets; 5y = 18183 as 3637 tickets; 10y = 18183 as 1818 tickets.

Or solve this system of equations:

1w + 2x + 5y + 10z = 90000

1w + 1x + 1y + 1z = 49999

1w + 6x + 0y + 0z = 90000

1w + 0x + 15y + 0z = 90000

for

w = 35453.1818; x = 9091.1364; y = 3636.4545; z = 1818.2273 .

Also, the third and fourth equations can be logically reformulated to make different counts of the ticket categories that add up to the required ticket total.