# A random graph

Probability Level 5

A graph is constructed iteratively according to the following algorithm. The graph starts as a single vertex. With probability $p$, the graph stops here. Otherwise, 3 new vertices are constructed and joined to this vertex. If we have a graph with more than one vertex, for each vertex created at the previous stage, 3 new vertices are construed and joined to it with probability $1 - p.$ We repeat this process until no new vertices are added. When the expected number of vertices in this graph is $100,$ the value of $p$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $a + b?$

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