We are going to present a very short (probabilistic) proof of the classical Dvoretzky theorem in the spirit of Figiel's (topological) proof. This approach gives rise of a new global parameter which is responsible for small deviation and small ball estimates. Furthermore, the dependence on epsilon we obtain in this way is the optimal one. A refined form of the random version of Dvoretzky's theorem (due to V. Milman) in terms of this parameter will also be discussed. The latter can be viewed as the almost isometric version of the Klartag-Vesrhynin result on the one-sided inclusion of Dvoretzky's theorem. (Joint work with G. Paouris). Speaker(s): Petros Valettas (University of Missouri)