In random events such a fair coin tosses, it is easy to predict over a long run of trials (flips) how many heads and how many tails should come up. However, very rarely does that number match the expected # exactly. For example, if you flip a fair coin $1000$ times you would expect about $500$ heads and about $500$ tails but you would be much more likely not to get exactly $500$ of either. So my question is what is the probability of getting exactly $500$ heads and $500$ tails out of $1000$ fair coin flips? It happens sometimes but not often so there should be a probability of it happening. If it is not the most probable outcome, then what is? $499$ + $501$, $498$ + $502$...?
I guess another way of asking this question is what % of the area under the bell curve for the graph of the possible outcomes for 1000 fair coin flips is exactly 500 heads? Is it something like a 5% chance for example?
Another example (for reference only) is random digits like in pi. If you take say $1$ million digits after the decimal point, you would expect $100,000$ of each of the digits $0$ thru $9$ but none of them in reality have exactly $100,000$ (although digit $8$ is very close with $99,985$ occurrences). It also seems that the larger the # of trials (coinflips or digits of pi for example), the harder it is to match the normal distribution exactly.