What is the probability of having an equal split between a high number and a low number when rolling a single dice multiple times?

Question

For example if you roll a dice 100 times and counting how often it's 1, 2, 3(low rolls) vs 4, 5, 6(high rolls) and repeating this experiment 10000 times, what fraction of time would the number of high rolls be equal to the number of low rolls?

Answer 1

So the chance of low rolls is the same as high rolls, $\frac12$ (assuming the dice is fair). Let $L$ be a lower number and $H$ be a higher number. Suppose there are $N$ rolls (for you $N=100\cdot10000=1000000$). Then there are $2^N$ possible permutations since there are $2$ possibilities ($H$ or $L$) for each roll. In the one we want, it has to have the same amount of $H$'s as $L$'s. There are $\dbinom{N}{0.5N}$ ways to do that for a total of $\frac{\dbinom{N}{0.5N}}{2^N}$ ways which is about $0.0008=0.08\%$ for $N=1000000.$