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A communication network is made of nodes conected with wire. The net sends packets in such a way that if one packet is located in an internal node $x$ (internal node is the one connected to more than one node), it chooses randomly the output node. The probability of going out through the node $y$ connected to $x$ is equal to $p_{xy}$, such that $\sum_{y} p_{xy}=1$. When the packet reaches an external node $X$, it remains there. $p_X$ denotes the probability of going to an external node, when we are connected to it.

We are thinking about calculating the probability $P(xX)$. That's to say, the probability that being the packet in an internal node $x$ it finishes in the external node $X$.

Solve the problem for a triangular net with nodes $a$,$b$,$c$ conected to the external nodes $A$,$B$,$C$,respectively. Solve it using linear equations that verify the different probabilities $P(xY)$.

If I want to the calculate probability $P(aA)$, I think that I have to solve a linear equation system with $P(aA)$, $P(bA)$ and $P(cA)$.

$P(aA)= P_A +P_A P_{ac}P_{ca} + P_A P_{ab}P_{ba} $. I don't know if I have to add the probabilites $P_A P_{ac}P_{cb}P_{ba}$ and $P_A P_{ab}P_{bc}P_{ca}$. Also if the packet goes from $a$ to $c$, instead of returning to $a$, $(P_{ac}P_{ca})$, it could go from $c$ to $C$.

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HINT (in case this is homework)

To answer your last question directly, yes you have to consider all complicated possibilities, like $abcbcbcbabacbaA$ etc. That's why that is not the way to go.

Instead, the key is to formulate this as a system of simultaneous equations. E.g. one of those equations is:

$$P(aA) = P_A + P_{ab} P(bA) + P_{ac} P(cA)$$

Why? Because from $a$, either you go to $A$ and win, or you go to $b$ in which case you now have prob $P(bA)$ of winning, or you go to $c$ and now have prob $P(cA)$ of winning. There are $3$ unknowns, $P(aA), P(bA), P(cA)$ and you need to find $3$ equations (one of them given above) and then solve. That accounts for $P(xA)$. Similarly you need other equation-systems for $P(xB)$ and $P(xC)$.

Can you finish from here?

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