We will be reviewing some simple random walk theorems. We will look at the amount of time that a SSRW is positive. If the random walk is at zero, it takes an even number of steps in either direction to return to zero.
We will look at the distribution when the random walk is positive. We can take an example with coin tosses. When we toss a coin, the probability that the number of heads equals k in 2n coin tosses is maximum when k = n . What we are saying is that we know for sure we can exceed the number of tails when the number of heads and the number of tails are equal. If we wanted to maximize the number for one outcome, we would have to choose the value 0 or 2n for the remainder.  Since we mentioned earlier that the probability a random walk reaches a certain count can be derived from the probabilities from the outcomes of the states from the distribution, we can express the random walk probabilities in terms of  the counts of the probabilities for the individual states.