What's the possibility of random guessing a youtube's video ID
Each YouTube id consists of 11 characters denoted by $v = [v_1, ··· ,
v_{11}]$. 
  The first $10$ characters of a valid id contain any of the characters
  in $S = \{0 − 9,\_, −, A − Z, a − z\}$, i.e., $v_i ∈ S (i =
1, ··· , 10)$. The last (11-th) character $v_{11}$ only comes from $T =
\{0, 4, 8, A, E, I,M, Q, U, Y, c, g, k, o, s, w\}$, namely, $v_{11} \in T 1$.
  The size of video id space is, therefore, $64^{10}\cdot16$. 
  
  We also observe that YouTube video ids are not generated in any sequence. Instead,
  YouTube picks an unused random id from this pool for each new
  video that is uploaded.
  
  Source: http://conferences.sigcomm.org/imc/2011/docs/p371.pdf


For instance, they show that YouTube video is randomly generated from the id
space and that the entire YouTube id space can be represented as $S = S^{10} × T$, where:


$S = \{0−9,\_, −, A−Z, a−z\}$ 
$T = \{0, 4, 8, A, E, I,M,Q, U, Y, c, g, k, o, s, w\}$.


Their estimation of the total number of youtube videos is $502\cdot10^6$.

Given these 2 information (video id space and estimation of the total number of videos) you can deduce the probability that, given an id of $11$ characters in $S^{10} × T$, it corresponds to the id of a valid video.