Use the time evolution operator where H is the Hamiltonian. (I'll leave it to you to find the Hamiltonian. Hint: It doesn't matter what kind of particle it is, if it's in a box the Hamiltonian can only have one form.)
The probability for which they occur can be found by
where is the eigenfunction for the eigenvalue .
Note: Properly speaking this should be but the wavefunction is real anyway, so I won't worry about this detail.
To do it more directly:
Write out the eigenfunction for the third energy level for the expanded box. Then the probability of the particle being in this state is
where is your original wavefunction.
The reason I am not certain about this is I can't think of what to do about the normalization at the moment. So your calculated in this manner is going to be off by a constant factor.