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449概率中等数值题medium

Memoryless Message Relay Chain

题目

A message must traverse a chain of relay nodes to reach its destination. Each node independently takes Geom(1/3)\operatorname{Geom}(1/3) attempts to successfully forward the message to the next node. However, on each attempt, there is an independent probability 1/51/5 that the node permanently fails, destroying the message. If the node fails, the message is lost. If the chain has 2 nodes, find:

(i) The probability the message reaches the destination (traverses both nodes).

(ii) The expected total number of attempts across both nodes, given the message reaches the destination.

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你的答案

(i)

(ii)