In this paper, we study simple symmetrical random walks and observe some seemingly unexpected results. Through some examples, we see that some seemingly unexpected results have a higher probability than we assumed. It is shown that the visits to the origin are not so frequent and that the probability that the particle has returned to the origin at some point in time is equal to the probability that it has not returned at all until that moment. We show that intuition is deceiving and that the previous part of the walk does not a ect the continuation (gambler’s ruin). We study random variables that have an arcsine distribution and monitor its properties. This shows us some interesting results that are surprising. It turns out that we don’t expect a lot of lead changes in the game and that there is a big chance one side spends most of the time in the lead. An interesting result is that if we fix that our walk ends at the origin, the distribution of spending time in the positive quadrant ceases to be an arcsine distribution and becomes unfornm. We find a good approximation for the probability at which point the random walk will end using the central limit theorem. Using simulations and graphs of the created simulations, we better understand the behavior of simple symmetric random walks and compare them with the expected results.