For all of the following answers look at the normalization of the wavefunction you are using.

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 different possible energy measurements are just the energy eigenvalues of the wavefunction. Hint: Apply the Hamiltonian to each separate term of the wavefunction.

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.

You know that the state is in the ground state. So the wavefunction is the ground state eigenfunction of the overall wavefunction. (What part of the given wavefunction is that?) Now apply time evolution to that eigenfunction.

You have the probability for each energy state to occur. So

To do it more directly:

I'm going to take my best shot at this now. I might post later if I change my mind after I've given it some more thought.

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.

-Dan