Probability of equally likely events

Question

I know that the probabilities of 2 equally-likely events is 0.5. Due to the fact that if i repeat the experiment for $N\rightarrow \infty$ the frequences of the 2 events are the same.

But, going to a deeper level, why this happens ?

Answer 1

Assuming your events are independent, we can model this using binomial distribution.

Let $X$ be the number of times event $A$ happens out of $n$ trial. Hence, $Y=n-X$ is the number of times event $B$ happens. We have $$X\sim\mathrm{B}(n,0.5)$$

Taking $n\to\infty$ is essentially taking expectation of $X$ and $Y$.

Proof of Expectation:\begin{align} \operatorname{E}\left[ X \right]&=\sum\limits_{r=0}^{n}{rP\left( X=r \right)} \\ & =\sum\limits_{r=0}^{n}{r\left( \begin{matrix} n \\ r \\ \end{matrix} \right){{p}^{r}}{{\left( 1-p \right)}^{n-r}}} \\ & =\sum\limits_{r=1}^{n}{\frac{r\cdot n!}{r!\left( n-r \right)!}{{p}^{r}}{{\left( 1-p \right)}^{n-r}}}\because {{\left. \frac{r\cdot n!}{r!\left( n-r \right)!}{{p}^{r}}{{\left( 1-p \right)}^{n-r}} \right|}_{r=0}}=0 \\ & =\sum\limits_{r=1}^{n}{\frac{n!}{\left( r-1 \right)!\left( n-r \right)!}{{p}^{r}}{{\left( 1-p \right)}^{n-r}}} \\ & =\sum\limits_{r=1}^{n}{\frac{n\cdot \left( n-1 \right)!}{\left( r-1 \right)!\left( \left( n-1 \right)-\left( r-1 \right) \right)!}p\cdot {{p}^{r-1}}{{\left( 1-p \right)}^{\left( n-1 \right)-\left( r-1 \right)}}} \\ & =np\sum\limits_{r=1}^{n}{\frac{\left( n-1 \right)!}{\left( r-1 \right)!\left( \left( n-1 \right)-\left( r-1 \right) \right)!}{{p}^{r-1}}{{\left( 1-p \right)}^{\left( n-1 \right)-\left( r-1 \right)}}} \\ & =np\sum\limits_{k=0}^{m}{\frac{m!}{m!\left( m-k \right)!}{{p}^{k}}{{\left( 1-p \right)}^{m-k}}};m=n-1,k=r-1 \\ & =np\sum\limits_{k=0}^{m}{\left( \begin{matrix} m \\ k \\ \end{matrix} \right){{p}^{k}}{{\left( 1-p \right)}^{m-k}}} \\ & =np\sum\limits_{k=0}^{m}{P\left( X=k \right)} \\ & =np\cdot 1 \\ & =np \end{align}

Thus, we arrive at $$E[X]=0.5n$$ and $$E[Y]=E[n-X]=n-E[X]=0.5n=E[X]$$

Answer 2

It is because of the Law of Large numbers

Also, you need to distinguish between theoretical probability and experimental probability: link 1 and link 2.