In quantum mechanics, the probability of obtaining the same measurement outcome after causing a wave-function collapse depends on the specific system and the initial state of the system.
When a measurement is made on a quantum system, the wave function of the system collapses to one of the possible eigenstates corresponding to the measured observable. The probability of obtaining a specific measurement outcome depends on the initial state of the system and the properties of the observable being measured.
If the system is prepared in an eigenstate of the observable being measured, then subsequent measurements of the same observable will always yield the same outcome. However, if the system is prepared in a superposition of multiple eigenstates, the subsequent measurement outcomes will be probabilistic.
The probability of obtaining the same measurement outcome again after the collapse of the wave function depends on the nature of the system and the specific measurement being performed. In general, if the system is prepared in the same initial state and the measurement is repeated under identical conditions, the probability of obtaining the same outcome will be high but not guaranteed.
It's important to note that the probabilistic nature of quantum mechanics arises from the inherent uncertainty and wave-like behavior of quantum systems. The exact probability distribution of measurement outcomes is determined by the wave function and the mathematical framework of quantum mechanics.