Difference between 0.5 of 5 heads out of 10 tosses and 0.5 of 50 heads out of 100 tosses? We know that the probability is 0.5 for 5 heads out of 10 tosses and likewise, 0.5 for 50 heads out of 100 tosses. However, we will say the probability of 0.5 from 100 coin tosses is preferable because we have tried more coin tosses but I don't know how to support my opinion.
Some says Hoeffding's Inequality but I am not sure that is going to help support my opinion since Hoeffding's Inequality looks like I could use that inequality for the number of data when I expect how small the error should be.[I am not sure if I am correct. I am still trying to understand the inequality. I am just saying it with how much I know, so far.]
Is there any theory to support that the probability of 0.5 from 100 tosses is better than the probability of 0.5 from 5 heads out of 10 tosses?
 A: One indication of the difference between (a) $x=5$ successes in $n=10$ tosses and (b)
$x = 50$ successes in $n = 100$ tosses lies in the different widths of the 95% confidence intervals (CIs) in the two cases:
Wald CIs. Even though the Wald Ci is not the most accurate kind of CI, it is perhaps
the most familiar, so I will start by making 95% Wald CIs for cases (a) and (b): [Roughly speaking, the 95% Wald CI extends 1.96 standard errors (mentioned in @Dave's Comment) on either side of the point estimate.]
#(a)
x = 5;  n = 10;  q = qnorm(c(.025,.975))
p.est = x/n
CI = p.est + q*sqrt(p.est*(1-p.est)/n);  CI
[1] 0.1901025 0.8098975

#(b)
x = 50;  n = 100;  q = qnorm(c(.025,.975))
p.est = x/n
CI = p.est + q*sqrt(p.est*(1-p.est)/n);  CI
[1] 0.4020018 0.5979982

So in (a) the 95% Wald CI is about $(0.19,0.81).$ of length about $0.62.$
By contrast, in (b) the CI is $(0.40, 0.60),$ of length about $0.20.$
Jeffreys CIs. A better style of frequentist CI is the
Jeffreys interval, which is based
on a Bayesian argument with a non-informative prior. This kind of interval
tends to have nearer the 'promised' 95% coverage probability--especially for
small $n.$ As shown by
the computations in R below, the two
95% Jeffreys CIs are about $(0.22,0.77)$ for (a) and $(0.40,0.60)$ for (b).
#a
qbeta(c(.025,.975), 5.5, 5.5)
[1] 0.2235287 0.7764713
#b
qbeta(c(.025,.975), 50.5, 50.5)
[1] 0.403174 0.596826

Note: With either style of confidence interval, the larger sample size leads to
a more precise estimate of the true success probability.
A: 
Is there any theory to support that the probability of 0.5 from 100 tosses is better than the probability of 0.5 from 5 heads out of 10 tosses?

It is not really "better" or "worse" - what you observe when you toss a coin 10 times or 100 times is not really a probability of getting heads per se, but rather a frequency of getting heads. However, this frequency approaches the true probability of getting heads in a fair coin toss when the total amount of tosses is increasing. This is why 100 tosses should serve as a better estimate of the true probability of getting heads than 10 tosses do, and 1000 tosses would be better than 100. Keep in mind that those are just estimates though; you may get 90 heads and 10 tails in your 100 tosses, which would not yield the probability of 0.5 for heads (nor for tails).
This is basically the frequentist definition of probability. Let $A$ be the "we got heads" event. Say we tossed the coin $n$ times, and we got heads $m_A$ times. Then the ratio $\frac{m_A}{n}$ is called frequency (I've also met the term "relative frequency") of $A$ - that is, of getting heads. This frequency, with increasing $n$, stabilizes itself near some non-random value that is actually a true probability of getting heads, according to the academic textbook by A.Kibzun (cannot quote it since it's in Russian). So if $\frac{m}{n} \rightarrow P(A)$ when $n \rightarrow \inf$, then obviously the greater $n$ you have, the closer you should be to getting the true probability.
Also:

According to the classical Bernoulli theorem, the relative frequency of an event $A$ in a sequence of independent trials converges (in probability) to the probability of that event.

From a paper by Vapnik and Chervonenkis (here's the translated version, this is the first phrase in the Introduction section).

The law of large numbers says that, for every single event, its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability

From wiki, I don't know the original source though, may not be as credible.
