Suppose that $x_1, x_2,...$ are independent copies of random variable $\xi$ having distribution $P(x=1)=P(x=-1)=\frac{1}{2}.$ Let $S_0=0$, $S_k=x_1+x_2+...+x_k$, $k \geq1.$ Let $P_{2k,2n}$ be the probability that during the time interval $[0,2n]$ the particle spends $2k$ units of time on the positive side.
1) Find all possible probabilities $P_{2k,4}$ using only definition.
2) Find all possible probabilities $P_{2k,8}$ using formula $$ P_{2k,2n}=u_{2k}*u_{2n-2k}.$$
So, problem is that I don't know what I have to do in 1). Maybe only then I can do also 2). So how to start and what to use in 1)?
1)
In time interval $[0,4]$ there are $2^4=16$ routes that can be taken. The routes are equiprobable.
Now you must pose yourself the question: "how many of these routes are such that the particle spends $2k$ units of time on the positive axis?" This for $k=0$, $k=1$ and $k=2$. It is just a matter of counting.
If there are $m_k$ such routes then: $$P_{2k,4}=\frac{m_k}{2^4}=\frac{m_k}{16}$$