Apparent contradiction with the definition of probability

#1
Hi everyone, here is my doubt: according to the definition of probability the probability that in a coin tossing the result be heads is 0.5 because in the limit of infinite trials (n==>infinity) half of times will be obtained heads. This is equivalent to say that in such a limit the probability of getting equal number of heads and tails is one and zero for any other combination. However, the de Moivre–Laplace theorem states that the binomial distribution tends to a normal one with SD √np(1-p) as n goes to infinity, where p is the succes probability that is 0.5 in this case. That would imply that the SD also tends to infinity! and not to zero as is expected from the aforementioned reasoning.

Thank you very much for your help
 

Dason

Ambassador to the humans
#2
There is a difference between "the number of heads" and "the proportion of heads".

Note that the SD of the proportion of heads is [math]\sqrt{p(1-p)/n}[/math] which goes to 0 as n goes to infinity.