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The tomographic invertable map of the Wigner function onto the positiveprobability distribution function is studied. Alternatives to the Schr\"odingerevolution equation and to the energy level equation written for the positiveprobability distribution are discussed. Instead of the transition probabilityamplitude (Feynman path integral) a transition probability is introduced.  A new formulation of the conventional quantum mechanics (without wavefunction and density matrix) based on the ``probability representation'' ofquantum states is given. An equation for the propagator in the new formulationof quantum mechanics is derived. Some paradoxes of quantum mechanics arereconsidered.Comment: Latex, 37 pages, to be published as lectures given at the Latin  American School of Physics, (XXXI ELAF, July-August, 1998, Mexico) in  Casta\~nos, O., Hacyan, S., Lopez-Pe\~na, R., (Eds.), AIP Conference  Proceedings (American Institute of Physics, New York, 1999

Conventional quantum mechanics without wave function and density matrix