What is an intuitive way to understand that if $X_1, \ldots X_n$ are iid, then $P(\max(X_1, \ldots, X_n) \leq x)= P(X_1 \leq x, \ldots ,X_n \leq x)$? I was wondering if there is an intuitive way to understand why it is that if  $X_1, \ldots X_n$ are iid, then $P(\max(X_1, \ldots, X_n) \leq x) = P(X_1 \leq x, \ldots X_n \leq x)$?
The standard explanation is that in order for the maximum to be less than something, every element must be less than that something as well. However, from a very rough intuitive standpoint, it seems that while $P(\max(X_1, \ldots, X_n) \leq x)$ cared only about the maximum, $P(X_1 \leq x, \ldots X_n \leq x)$ now cares about each and every variable, so it seems that we are working with more information in the latter. Basically, it feels as if $P(X_1 \leq x, \ldots X_n \leq x)$ has more "going on" than $P(\max(X_1, \ldots, X_n) \leq x)$. Is there a nice way to think about this clearly? Thanks!
 A: It has nothing to do with iid, or probability for that matter.
"Each person in the room is at most 80 years old" is equivalent to "The oldest person in the room is at most 80 years old".
A: 
However, from a very rough intuitive standpoint, it seems that while $\mathsf P(\max(X_1 ,…,X_n )≤x)$ cared only about the maximum, $\mathsf P(X_1 ≤x,…,X_n ≤x)$ now cares about each and every variable

While the first probability is only concerned about the maximum, the maximum itself is concerned about the values of the individual variables.
The maximum of a list of variables is less than a value if and only if each and every variable in the list is less than that value.   $\big\{\max\limits_{k=1}^n\{X_k\}\leq x\big\}$ and $\bigcap\limits_{k=1}^n\{X_k\leq x\}$ are equivalent, as they describe the same event.
A: since $X_{i}$'s are identical, we can write this $P(X_{1}\leq X_{2})=P(X_{1}\leq X_{i})$ for any i=2,...,n. It does not matter which value of i you consider in above equality.and, since X_{i}$'s are iid then we can intuitively see why your equality holds.
