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?