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Let's start from simplified case, that is trivial to solve, then I wil explain the problem that I have (it's not a homework of any kind or anything, I just want to know expected value of the rewards I can win in certain game (I will call the reward in dollars, but that is just a placeholder, you can call it just an resource unit instead)).


Let's say that we have some lottery tickets, where we can win different things, but only two of possible rewards are of our interest:

  • We have $p$ chance for winning highest reward $X$, let's say $X = 300k$ dollars (\$300000).

  • We have $q$ chance that ticket will reward us with another ticket

Each ticket is independent of other tickets and we can win the main reward as many times as many tickets we have (if we are "lucky" enough). In other words, possible rewards and their probability distribution never changes, no matter what we won before.


For example, let's say that we have $10$ tickets, $p=0.0034 $ and $ q=0.01$.
For each ticket, we have:

  • $p$ chance for winning $X$ ($300k$ dollars),
  • $pq$ chance for winning ticket, then $X$ in that ticket,
  • $pq^2$ chance for winning ticket, which contains ticket with $X$ ticket,

and so on.

Thus, ignoring all the other rewards that have $1-p-q$ chance of being drawn, our ticket is worth

$$X\cdot (p+pq+pq^2+pq^3...) = X \cdot \frac{p}{1-q} = 1030.303030...$$ So, out of $10$ tickets, we can expect about $10303$ dollars (the numbers are made up, just an example).



Going to the point:

Let's say that we have a lottery ticket that have:

  • $1-p-q$ chance for stuff irrelevant for us,
  • $p$ chance for main reward worth $X$,
  • $q$ chance for $N$ tickets.

Now, I don't know how to approach it. It's trivial with re-rolling a single ticket, but if each ticket can give you $N$ next tickets, then every single of these tickets can again be either the winning ticket or the one that gives you another $N$ tickets... I don't know how to approach it.

How do I solve it, to find an expected value of such ticket? Do I have to find some recurrence to simplify the problem and make it solvable?

I will be glad for help with solving it or some real advices.


Extra question:

Can we easily generalize it even further, to each ticket having:

  • $p$ chance to win $X$,
  • $q_{k}$ chance to win $k$ more tickets, $k=1,\dots,s$,
  • $1-p-q_1-...-q_s$ chance to win something else (irrelevant for us)?

How would we find expected value of the ticket in such case?

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  • $\begingroup$ Sorry for my chaotic writing style. If you think that I should change something in the question, feel free to comment or edit. If no one will answer in 2 days, I think I will try adding some small bounty, even though I don't have too much of reputation :-) $\endgroup$
    – Kusavil
    Dec 21 '19 at 20:51
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Let $E$ be the expected value of a ticket. Then the expected value of $N$ tickets is $NE$. We have $$ E=pX+ qNE$$ so that $$E=\frac{pX}{1-Nq}$$

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  • $\begingroup$ Wow, so simple! Thank You! Looks like not studying or "doing math" for a long time makes one's brain numb, I really have to fix this soon :-) $\endgroup$
    – Kusavil
    Dec 22 '19 at 16:22

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