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For a cubic box the energy of a particle is given by:

$$ E = \frac{\pi^2 \hbar^2}{2mL^2} [n_1^2+n_2^2+n_3^2]$$

It is possible that $E$ has the same value for different combinations of $n_1, n_2$ and $n_3$.

I have read that when there are states with the same energy, this energy level is said to be degenerated.

Does this mean that the probability to get this level is $0$?

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Degeneracy means, an energy-level can be occupied by different combinations of quantum number $n_i$. So you are correct in this point. But this means, there are multiple eigenstates with the same value, so their probability is the same for every eigenstate and not $0$.

A more thorough explanation can be found here.

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