Expected number of events in a simple birth-death process.

#1
I have a simple birth-death process with parameters, beta=4/5v and the initial population size is 6. I have calculated that the birth rate= 4/9 and the death rate=5/9 but is there a way of calculating how many events (births and deaths) there will be until this population dies out?
I have a feeling it may use Gambler's ruin methods.
 
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