I've been trying for a few hours now to wrap my brain around, what at first, seemed like a simple concept. I've tried so many different things and ways of constructing scenarios so I can figure out how to calculate this, but I just can't seem to get it right
Here's the scenario:
Imagine there are 25 different TYPES of "prizes" each worth a monetary value. The total quantity of all the prizes is infinite.
Type. Value
1. $0.5
2. $0.5
3. $1
4. $1
5. $1
6. $1
7. $2
8. $2
9. $2
10. $4
11. $5
12. $5
13. $7
14. $8
15. $10
16. $10
17. $12
18. $15
19. $15
20. $20
21. $20
22. $20
23. $25
24. $30
25. $100
Now let's say that you ran a "casino" and someone could pay $2 in return for one of the prizes. What winning-percentage would the "casino owner" have to set for each type of prize so that in the long run, the buyer would walk away even? The buyer should lose as much money as they gain.
In other words, if the buyer has lost \$98 due to lots of bad luck, the \$100 prize should be the next prize to be won.
This is also taking place in a "perfect world"… The smallest prizes should have the largest percentages.
At first, I just thought about how many \$2 payments it would take to make your money back for each value (50 payments to win the \$100 prize). So that means a 2% chance for the \$100 prize? But when I did that for all the rest, it was clear to me that was the wrong approach since it was a 100% chance for the \$2 prize.
How do I calculate the percentage chance to win each prize so the buyer walks away with the "same" amount of money after a gazillion plays? Thanks for your time. 🙂
Best Answer
I'll post this answer assuming that it's possible to win no prize, which seems reasonable.
Of course, there's no unique answer to this question, since for example, you could simply always give back the $8^{th}$ prize. Still, I took an approach that tries to go around this problem of non-uniqueness.
Let's use for that the concept of RTP. We can force the game to make every prize has the same RTP and that all together the game has an RTP of 100%. What I mean by every prize having the same RTP is that each price, taking into account how often it appears, contribute the same to the total RTP. So for each prize the formula $prize\times P(prize)$ remains constant. That means that $RTP=4\%$.
Now let's find what should be $P(prize_n)$ where $prize_n$ is winning the $n^{th}$ prize.
$$P(prize_n)=2\times\frac{4\%}{prize\ value} $$
This gives the following table of probabilities
Probabilities with * had been rounded so if you have to implement it, I'd recommend you to use frecuencies ($\frac{1}{probability}$), that are not rounded. That's why I added the Frecuency column as well. Frecuencies are telling you that you should see that prize once for every time you'd played that number of games. So, you should see one prize 7 every 25 prize (that's the expected value), or after 100 games you should see 16 prizes 1.
The solution for your problem if you always win a prize requires some computation.
Let $p_i$ be the probability of wining prize $i$. Let $v_i$ the value of the przie $i$. You have the following system.
$$ \sum_{i=1}^{25} p_i=1\\ \sum_{i=1}^{25} p_iv_i=2\\ p_i>p_{i+1}\ \ \forall i\in [1,24] $$
Last condition wouldn't be needed but it's for making it more beautiful, every prize is more probable the smaller it is.
Here you have 26 equations (actually 2 equations and 24 inequations) that can solve through computer.