modified random walk problem

Question

so there is a problem of a machine which has a screen that can show integers. starting at $0$ at each time step the machine either increases the number by $1$ or decrease it by $1$ (the probability of these two actions are equal). if we know the machine have worked for T rounds, what is the probability of seeing $0$ at least one time on the screen during this time interval. for example for $T=4$ these streams can be produced by the machine:

1. [1, 1, 1, 1]
2. [1, 1, 1, -1]
3. [1, 1, -1, 1]
4. [1, 1, -1, -1]
5. [1, -1, 1, 1]
6. [1, -1, 1, -1]
7. [1, -1, -1, 1]
8. [1, -1, -1, -1]
9. [-1, 1, 1, 1]
10. [-1, 1, 1, -1]
11. [-1, 1, -1, 1]
12. [-1, 1, -1, -1]
13. [-1, -1, 1, 1]
14. [-1, -1, 1, -1]
15. [-1, -1, -1, 1]
16. [-1, -1, -1, -1]

the acceptable streams are:
[1, 1, -1, -1]
[1, -1, 1, 1]
[1, -1, 1, -1]
[1, -1, -1, 1]
[1, -1, -1, -1]
[-1, 1, 1, 1]
[-1, 1, 1, -1]
[-1, 1, -1, 1]
[-1, 1, -1, -1]
[-1, -1, 1, 1]

so the probability will be $\frac{10}{16}$