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We consider an initial Eve-population and a population of neutral mutants,such that the total population dies out in finite time. We describe theevolution of the Eve-population and the total population with continuous statebranching processes, and the neutral mutation procedure can be seen as animmigration process with intensity proportional to the size of the population.First we establish a Williams' decomposition of the genealogy of the totalpopulation given by a continuous random tree, according to the ancestrallineage of the last individual alive. This allows us give a closed formula forthe probability of simultaneous extinction of the Eve-population and the totalpopulation.

Williams' decomposition of the L\'evy continuous random tree and simultaneous extinction probability for populations with neutral mutations