hello=To create a formula that represents the condition where, when dividing the Powerball lottery into 4 groups according to the parity of the digits (PP, II, IP, PI), one of the groups has zero or one number and another group has 2 to 4 numbers, we can think of something like this:
1. Definitions of the groups:
• PP: first digit even, second digit even
• II: first digit odd, second digit odd
• IP: first digit odd, second digit even
• PI: first digit even, second digit odd
2. Let xPP,xII,xIP,xPI be the number of numbers drawn in each group.
3. The desired condition is:
• One group xgx_gxg (where g∈{PP,II,IP,PI}g \in \{PP, II, IP, PI\}g∈{PP,II,IP,PI}) has 000 or 111 numbers,
• Another group has between 222 and 444 numbers,
• The other two carry the remaining numbers.
4. To express this, we formulate:
∃g1,g2∈{PP,II,IP,PI},g1≠g2:(xg1=0 or 1)∧(2≤xg2≤4)\exists g_1, g_2 \in \{PP, II, IP, PI\}, g_1 \neq g_2 : (x_{g_1} = 0 \text{ or } 1) \land (2 \leq x_{g_2} \leq 4)∃g1,g2∈{PP,II,IP,PI},g1=g2:(xg1=0 or 1)∧(2≤xg2≤4)
Furthermore, the total sum must be equal to the total number of numbers drawn, which in Powerball is 555 numbers (not counting the ball). (Special):
xPP+xII+xIP+xPI=5x_{PP} + x_{II} + x_{IP} + x_{PI} = 5xPP+xII+xIP+xPI=5
5. Since one group "carries more" and another "carries less," the formula emphasizes that the numbers are unevenly distributed, but the total quantity is fixed.
6. If you want the formula to return true only under these conditions, you can structure the formula in code or mathematical logic checking the values of xgx_gxg with these restrictions.
In summary, the model is:
• For the counters xPP, xII, xIP, xPI, the number of numbers in each group,
• Valid if there is a group with 0 or 1 number,
• And there is a group with between 2 and 4 numbers,
• Totaling 5 numbers added in the four groups.
If needed, I can also help with the implementation of this in Excel, VBA, or another language.
If desired, I can detail the logic to filter or create functions that evaluate this distribution of numbers in the Powerball according to these criteria.
1. https://pypi.org/project/powerball/
2. https://lotterycodex.com/lottery-formula/
3. https://floridalottery.com/content/dam/floridalottery-web/files/game-reports/powerball/pwrball-odds.pdf
4. https://www.powerball.com/powerball-prize-chart
5. https://www.durangobill.com/PowerballOdds.html
6. https://lotterycodex.com/how-to-win-powerball/
7. https://www.palottery.pa.gov/Draw-Games/Powerball.aspx
8. http://www.senecanewsdaily.com/images/Powerball-Jackpot-Distribution.pdf
9. https://www.reddit.com/r/theydidthemath/comments/1n7wlit/what_am_i_missing_about_powerball_oddsrequest/
10. https://www.powerball.net/statistics