Limit Theorem applicability A basketball player throws 100 times. On the first throw the ball is always in, on the second - always out. For all consequent throws the probability of hitting is equal to percentage of all successful throws divided by total number of throws done.
What is the probability the he will score total 50 balls out of 100?
This is not a homework, I'm just anxious about the solution of this problem.
My ruminations:
There are random events $E_1, E_2, \dots, E_n$. $E_1$ always happens, $E_2$ never. For any $E_n$ with $n > 2$:
$p(E_n) = \frac{\sum_{k=1}^{i-1}{x_k}}{i-1}$
${x}$ a sequence of random variables, such that: $x_i = 1$ if event is happened, $0$ otherwise.
So,
$P (\sum_{i=1}^{100}{x_i} = 50) = ?$
Is it true that this problem can be solved with Limit Theorem? Consequent events are dependent on outcomes of previous; can the Limit Theorem handle that? Could give me a pointer on how can I solve this problem?

The probability is $1/99$.
Denote by $P(a,b)$ the probability that after $a+b$ shots, exactly $a$ were in and $b$ were out. Then $P(1,1)=1$ and
$P(a,b) = \frac{a-1}{a+b-1} P(a-1,b) + \frac{b-1}{a+b-1} P(a,b-1)$
We prove by induction on $a+b$ that $P(a,b) = 1/(a+b-1)$. When $a+b = 2$, this is true. Otherwise
$P(a,b) = \frac{a+b-2}{a+b-1} \cdot \frac{1}{a+b-2} = \frac{1}{a+b-1}$
