Is "a fair coin being tossed $n$ times" the same as "$n$ fair coins being tossed once"? This is possibly a follow-up question to this one:
different probability spaces for $P(X=k)=\binom{n}{k}p^k\big(1-p\big)^{ n-k}$?
Consider the two models in the title:


*

*a fair coin being tossed $n$ times

*$n$ fair coins being tossed once


and calculate the probability in each model that "head" appear(s) $k~ (0\leq k\leq n)$ times. Then one may come up with the same answer that
$$
P(\text{"head" appear(s)} ~k~  \text{times}) = \binom{n}{k}p^k\big(1-p\big)^{n-k}
$$
However, the first one can be regarded as a random process, where the underlying probability space is $\Omega = \{0,1\}$ ($1$ denotes "head" and $0$ for "tail") and the time set $T=\{1,2,\cdots,n\}$. While in the second one, the underlying probability space is $\Omega = \{0,1\}^n$. 
Here are my questions:


*

*How can I come up with the same formula with these two different points of view?

*Are these two models essentially the same?

 A: The models are essentially the same.  I think this automatically answers your first question as well.
You can see the two as trading a space dimension for a time dimension.
A: Both models are basically a way to put a probability in $\{0,1\}^n$.
Usually you will be given a probability distribution in $\{0,1\}$,
and try to extend it to a probability in $\{0,1\}^n$, according
to some extra assumption.
If one experiment (tossing a coin)
does not influence the other (tossing it again, or tossing another coin),
then, you will have the model you describe.
The point is that when you talk about a random process,
usually you are allowing that the result of an experiment (toss a coin)
might influence the result of the next (toss it again).
Changing this condition, you might get a different probability
distribution in $\{0,1\}^n$.
For example, it might be assumed that when the outcome is $1$,
then the probabilities for the next outcome are flipped,
that is $p$ becomes $1-p$.
A more concrete example is the probability of a certain letter
appearing in a text. After a consonant, it will be likely that
the next letter will be a vogue. After a "p", we will not be likely
to get a "x" or a "w".
