Repeated coin flips probability Assume in an experiment, one flips a coin $L$ times. This experiment is repeated N times. We represent these in a table with $N$ rows and $L$ columns with order.
So a column is defined at the position of the $k$'th flip ($1 \le \forall k \le L$) among all experiments.
If the head probability is $P_h$, then what is the probability of the following events:
1- "No two experiments share a heads in a column"
2- " In at least one column, all experiences are heads."
Also I would like to know what kind of random process can model this.
Hint: I guess this can be addressed as a special case of the birthday problem. Each experiment can be seen as a person while the "head" event is a kind of "birthday". So we have $N$ individuals to share / not share these events.
 A: The probabilities may be calculated, assuming all results are independent binomial outcomes.  Here $P_h$ is probability of a single coin toss resulting in heads.
Although framed as $N$ repetitions (the rows) of $L$ coin tosses, the first part lends itself more easily to considering $L$ repetitions (the columns) of $N$ coin tosses.  We are asked the chances that no column will contain more than one head entry, equiv. that we $L$ times get such an outcome, i.e. $Pr(\text{at most one head in }N\text{ tosses})^L$.  The probability of getting at most one head in $N$ attempts is $(1-P_h)^N + N P_h (1-P_h)^{N-1}$ by the usual binomial expansion.  So raise this to the power $L$ to answer the first part.
The second part also lends itself to considering the columns as independent trials.  The chance of getting all heads in a fixed column is $P_h^N$, so the probability that at least one column has all heads is the complement of having no columns with all heads:
$$ Pr(\text{at least one column all heads}) = 1 - (1-P_h^N)^L $$
Set to Community Wiki, so anyone who spots a mistake or lacuna should feel free to fix it.
