anybody knows how to derive the simulataneous equations for conditional expectation from the engel diagram in the case of tossing a coin...i.e. for example e(w) be the expected number of trials needed to see w for the first time...=> so e(HHH) be the expected number of trials needed to see HHH for the first time (just like that)

in this case the simultaneous equations is e=p*1+q(1+e)

anyone know how we get this simultaneous equations?

thanks