Assume we have a lottery with payouts $(2,3,5)$. So if you buy a ticket you can win a pot which will payout your ticket price multiplied by one of those numbers.
The organizer expects a margin profit of $4\%$ from all tickets. So if the player plays with $1\$$ the mathematical expectation of outcomes will be $0.96$.
In my previous question I asked almost the same question thinking that my approach calculating the probability of each payout was right. I wanted the probability of each payout to be proportional to the payout $P_i \propto \frac1{x_i}$ (where $P_i$ is the probability of winning the $i$ payout and $x_i$ is the value of that payout). So I got an answer with these probabilities $(0.192,0.096,0.0576)$ and this explanation $$ \begin{align} \sum P_i \cdot x_i &= 0.192 \cdot 2 + 0.096 \cdot 3 + 0.0576 \cdot 5 \\ &= 0.96 \end{align} $$ Can someone please explain or help me to understand how these probabilities can be counted and the idea behind that.
My previous method which I used and assume was wrong looked like this.
If $P_i$ is the probability of payoff $i$ and there are $N$ total positive payoffs that $P_ii = 0.96/N$. Then you get the formula that $P_i = \frac{0.96}{iN}$, which would look like this in my question
$P_2 = \frac{0.96}{2*3}$ where $3$ is the number of positive payouts.
Thank you