  We apply the recently proposed quantum probing protocols with an unknown
system-probe coupling to probe the convex coefficients in mixtures of commuting
states. By using two reference states instead of one as originally suggested,
we are able to probe both lower and upper bounds for the convex coefficient. We
perform extensive analysis for the roles of the parameters characterizing the
double peaked Gaussian frequency spectrum in the Markovian-to-non-Markovian
transition of the polarization dynamics of a single photon. We apply the
probing of the convex coefficient to the transition-inducing frequency
parameter and show that the non-Markovianity of the polarization dynamics can
be confirmed with a single snapshot measurement of the polarization qubit
performed at unknown time and even with unknown coupling. We also show how the
protocol can identify Markovian and non-Markovian time intervals in the
dynamics. The results are validated with single photon experiments.
