To find the probability that a measurement of the z-component of spin of the spin-1 particle is zero, we need to consider the possible outcomes and their probabilities.

Given information:
- Two particles with spin -3/2 and spin -1 are at rest in a box.
- Total spin is measured to be 5/2.
- Total z-component of spin is found to be 3/2.

Total spin and its z-component:
The total spin of the system is given by adding the individual spins:

Total spin = spin1 + spin2

Given that the total spin is 5/2, we can write the equation as:

5/2 = -3/2 + spin2

Solving for spin2, we get:

spin2 = 5/2 + 3/2 = 8/2 = 4/2 = 2

This tells us that the spin of the spin-2 particle is 2.

Possible outcomes for the z-component of spin:
The z-component of spin can take values from -spin to +spin in integer steps. In this case, since the spin of the spin-1 particle is 1, the possible outcomes for its z-component are -1, 0, and 1.

Calculating probabilities:
To find the probability for a specific outcome, we need to consider the possible outcomes and their probabilities. Since the total z-component of spin is found to be 3/2, we know that the sum of the z-components of the two particles must be 3/2.

Let's consider the three possible outcomes for the z-component of the spin-1 particle:

1. If the z-component of the spin-1 particle is -1, the z-component of the spin-3/2 particle must be 3/2 - (-1) = 5/2. However, the spin-3/2 particle only has z-component values of -3/2, -1/2, 1/2, and 3/2. Therefore, this outcome is not possible.

2. If the z-component of the spin-1 particle is 0, the z-component of the spin-3/2 particle must be 3/2 - 0 = 3/2. This is one of the possible outcomes.

3. If the z-component of the spin-1 particle is 1, the z-component of the spin-3/2 particle must be 3/2 - 1 = 1/2. However, the spin-3/2 particle only has z-component values of -3/2, -1/2, 1/2, and 3/2. Therefore, this outcome is not possible.

Since there is only one possible outcome (z-component of spin-1 particle is 0) out of the three considered, the probability is 1/3.

Therefore, the probability that a measurement of the z-component of spin of the spin-1 particle is found to be zero is 1/3.