I have a standard Poisson process $N(t)$ with arrival rate $\lambda$ per second. Each time an arrival occurs, a fair coin is flipped and a counter is increased or decreased by $1$ depending on the result. If we let $M(t)$ indicate the counter value at time $t$ and let $M(0) = 0$ (just like $N(0) = 0)$, at time $t$, if we knew that $N(t) = n$, then the value of the counter $M(t)$ could be understood as
$ \# \text{(successes)} - \# \text{(failures)}$
in $n$ Bernoulli trials with success probability $\frac{1}{2}$ . Using this logic, i need to find the conditional PMF $\mathrm{Pr}[M(t) = m\mid N(t) = n]$ and the expression for the PMF of $M(t)$.
I am having trouble on how to approach the problem and how # successes - # failures could be understood in finding the PMF. Any help would be greatly appreciated. Thanks!