# How to solve this probability related equation?

I am trying to wrap my head around probabilities, currently trying to solve puzzles related to the topic. One puzzle involves two people throwing a dice one after the other. Whoever gets a six first wins, the question is "What is the probability that the person throwing first wins?". Since it is a tutorial puzzle, i am given the information that it is not necessary to calculate an infinite sum to solve it, but instead solve for $$p$$ in the following equation: $$p = {1 \over 6} + \left({5 \over 6}\right) ^ 2 p$$ How can i solve the equation for $$p$$, and is it always possible to solve the generalized equation $$p = {1 \over x} + \left({x-1 \over x}\right) ^ 2 p$$ for $$p$$, with $$x \in \mathbb{N}$$ and $$x \neq 0$$?

New contributor
Zweitopf is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
• Not sure what you are asking. If $p=A+B\times p$ for some $B\neq 1$ and $A\neq 0$ then $(1-B)p=A\implies p=\frac A{1-B}$. Is that what you are asking?
– lulu
Commented Aug 1 at 13:15
• @lulu Yes, that answers my question. Commented Aug 1 at 13:24

The person who throws first will win iff [his first throw is a six] or [his first throw is not a six and his opponent (who is now in the position of being the first to throw) looses].

So if $$p$$ denotes the probability that the person who throws first wins then we have the equality:$$p=\frac16+\frac56(1-p)$$which is easy to solve and makes calculation of any infinite sums redundant.

If you had written the equation properly, that would also have given you the solution without recourse to computing an infinite sum

$$p = \frac16 + \left(\frac56\right)^2p$$

means that either you win on first toss or you and your opponnt both lose on respective turns, which brings you back to square $$1$$, i.e. start

It is then a routine matter to get
$$\frac{11}{36}p = \frac{6}{36} \Rightarrow p = \frac6{11}$$

Of course, @drhab's approach is simpler than that in your tutorial