However, is it not also backed by arithmetic that if you make 500 flips or rolls, for example, at 50% chance of winning, even if the results would not exactly balance out against each other, it will definitely be not far from each other, or the discrepancy will be at the minimal level, right? It is close to impossible that out of 500 rolls, 400 turns out red and only 100 turns out green.
Assuming we are talking about something which has a 50/50 chance of happening (such as flipping a fair coin) here:
You can work out exactly how "close to impossible" it is. The probability of flipping at least 400 heads out of 500 flips is 8.29815×10
-44, so yes, very close to impossible. You can use standard deviations to see the probability of the results not being far from each other, as you put it. At 2 standard deviations from the mean, 95% of the time you would end up in the range of 228 - 272. At 3 standard deviations, 99.7% of the time you would end up in the range of 217 - 283.
The mistake I think you are making here is conflating "all rolls" with "individual rolls". Let's make the numbers smaller to make it easier to follow. Let's say I flip a coin three times. There are exactly 8 possible outcomes:
HHH
HHT
HTH
HTT
TTT
TTHTHT
THH
If you look at all 8 results, you will see there is only 1 which has no heads, so flipping three tails has a probability of 1 in 8, and flipping at least 1 head has a probability of 7 in 8. Now, lets say I have just flipped the coin twice, and flipped two tails. Look at the two I have made bold. I am going to flip a third time. The other 6 out of 8 options are irrelevant at this point, because the probability of any of them happening is now 0, because I have just flipped two tails in a row. That means my next flip only has two possible outcomes - TTT or TTH, therefore a chance of 1 in 2. What happened before is irrelevant.
Now lets go go back to the start of this example and look at my two possible final outcomes - TTT or TTH. From the very start, before I flipped anything, they are both as likely as each other. Both have a 1 in 8 chance of happening. This is the same when we scale things up. Flipping 19 heads in a row followed by another head is exactly as probable as flipping 19 heads in a row followed by a tails.