Does random walking have a memory? I do not know if this is a good question, but I have not found an answer for it anywhere, does a binomial random variable have memory, [simple logic says it can have no memory, but statistics show that the more times we toss a fair coin, the more likely it is to fall on heads around fifty percent of the time, and ostensibly this means that the coin has a memory and is affected by what has happened to it in previous times]
Thank you very much for any answer or comment.
 A: By definition, if your observations are independently drawn, they are memoryless in the sense that $P(\text{new event}|\text{past events}) = P(\text{new event})$. In the context of a binomial random variable, you can think of that as $n$ independent bernoulli's, and so the same concept will hold: by definition of binomial, the $k$-th experiment is independently of past experiments.
You may be confusing memory with issues somewhat similar in flavor to the concept of regression toward the mean. It's not that each individual coin flip (of an equally weighted coin) is aware of the past coin flips and is basing where it lands depending on those past flips to 'even things out', but rather that as you gather more observations, the sample mean is more likely to be closer to the true mean (in this case, the probability of flipping heads). When you have few observations, which happens at the start of a sequence of observations, you'll notice the average may seem quite off from the true average for a single sequence of observations, but as you gather more and more observations, it seems to settle to the true mean. One way of convincing yourself that this is not due to memory is to note that had you taken any similar small number of observations from the end of your sequence of data, you'd more likely observe a similar issue with the mean being far off from the true mean (however, if you even did this over and over, that average would also settle to the true mean!). This is more a reflection of concepts such as the law of large numbers or the fact that for many distributions, as your sample size grows, your estimate of the mean becomes more precise (in some sense, this is exactly what the CLT also says).
Edit:
To respond to your comment, the point is that the overall mean will (eventually) converge to the true mean, but this says nothing about individual coin tosses. I think the confusion may also stem from confusing a random variable with realized observations. To illustrate, look at the below figures I build (code at end). I sampled 1000 binomial (with $p=.5$). On the left, I plot the mean of the first x observations for each x, and on the right, I plot the mean of the last 20 observations for every 20 xs. On the left, you see the law of large numbers kick in: in the long run, the total mean converges to the true probability, but look at the right: there is no pattern at all... sometimes the mean of the next 20 observations is similar to the last one, sometimes totally different!
What you may instead see is that if for a given number of observations you happen to randomly observe lots of tails, then if you did it again, just probabilistically, it's unlikely you will again observe lots of tails, since that event is unlikely, but that's not memory, that's just probability. Similarly, if you play the lottery and win, and then next 10 times you play you don't win, its not memory driving that result, that's just probability telling you you won't win, but you're comparing it to an observed win that is extremely unlikely.

Code:
set.seed(20)
n = 1000

samp = rbinom(n,1,.5)

tot_mean = sapply(1:n, function(x) mean(samp[1:x]))
bin_mean = sapply(1:n, function(x) ifelse(x %% 20 == 0 ,mean(samp[x:(x+20)]),NA))

#now plot both side by side..
require(data.table)
require(ggplot2)
require(ggpubr)

dt = data.table("num" = 1:n,
                "mean_up_to_x" = tot_mean,
                "mean_past_20" = bin_mean)
g1 = ggplot() +
  geom_point(aes(x = num, y = mean_up_to_x), data = dt) + 
  geom_hline(yintercept=.5, linetype="dashed", color = "red")

g2 = ggplot() +
  geom_point(aes(x = num, y = mean_past_20), data = dt) +
  geom_hline(yintercept=.5, linetype="dashed", color = "red")

g <- ggarrange(g1,g2, ncol = 2)

A: Have a look at the Gambler’s fallacy, that should answer your question.
In a nutshell, no, independent events don’t have a memory. They’re independent! If I’m flipping a coin and get a run of heads, the coin doesn’t remember this and then start to produce more tails to compensate. Each flip is still exactly 50/50 (assuming fair, of course).
What that means is that there will be a tendency “regression to the mean”. If we ignore the early string of heads, and just think about the future N flips, the coin should go on produce an equal amount (N/2) of heads and tails.
If we now reintroduce the extra heads into our thinking, it’s true that there will be a small excess amount of heads. But as N tends to infinity, this excess will become more and more negligible and the ratio of heads to tails will tend to N/2. This is regression to the mean.
Of course it’s perfectly possible to get an exactly equal amount of excess tails at some point that will exactly compensate the excess heads - but that’ll be purely luck. And then, on the very next throw you’ll have a single excess of either heads or tails. Of course, as N is now very big, that single excess is negligible. Or you might get too many tails and then have an excess of them - which will also become negligible in the long run.
In simpler words, an excess of either heads or tails can (and will) happen in any finite run. But in a very long run the excess will negligible. A blip into an excess of something is just a bit of transient “noise” that will be swallowed in the limit of very large N.
