In quantum mechanics, probabilities are derived from the mathematical framework known as wavefunctions or state vectors. These wavefunctions describe the quantum state of a system and evolve over time according to the Schrödinger equation or other appropriate equations depending on the specific situation.
When we make observations or measurements in quantum mechanics, the wavefunction collapses to one of the possible eigenstates of the observable being measured. The probability of obtaining a particular measurement outcome is given by the square of the absolute value of the corresponding coefficient in the wavefunction expansion, which is often represented by a complex probability amplitude.
In this sense, probabilities in quantum mechanics are inherently probabilistic and not determined by statistical distributions of previous observations. Instead, they arise from the nature of quantum states and the act of measurement itself. The probabilities reflect the likelihood of obtaining specific measurement results when repeated measurements are made on identically prepared systems.
It's important to note that the interpretation and understanding of quantum mechanics are subjects of ongoing debate among physicists, and different interpretations exist, such as the Copenhagen interpretation, many-worlds interpretation, pilot-wave theory, and more. These interpretations provide various perspectives on the nature of probabilities in quantum mechanics.