To solve this problem, we will use the microcanonical ensemble in statistical mechanics, which deals with systems of fixed energy, volume, and number of particles. In this ensemble, all accessible microstates are assumed to be equally probable. Let's denote the total number of particles as N = N1 + N2 and the total energy as E = E1 + E2. We want to find the probability that the particles will be distributed equally between the two compartments, i.e., N1 = N2 = N/2.First, we need to calculate the total number of microstates _total for the system. This can be done using the formula:_total = 2V/h^3 ^3N/2 * 2mE/3N ^3N/2 * N! / N1! * N2! where V is the volume of each compartment, h is the Planck's constant, m is the mass of the particles, and E is the total energy.Next, we need to calculate the number of microstates _equal when the particles are distributed equally between the two compartments. This can be done using the same formula, but with N1 = N2 = N/2:_equal = 2V/h^3 ^3N/2 * 2mE/3N ^3N/2 * N! / N/2 ! * N/2 ! Now, we can find the probability P of the particles being distributed equally between the two compartments by dividing the number of microstates with equal distribution _equal by the total number of microstates _total :P = _equal / _totalAfter canceling out the common terms, we get:P = N! / N/2 ! * N/2 ! / N! / N1! * N2! P = N1! * N2! / N/2 ! * N/2 ! This is the probability that the molecules will eventually be distributed equally between the two compartments. Note that this probability depends on the initial values of N1 and N2, and it will be higher when the initial distribution is closer to equal.