Some probabilistic Analysis

Let's have x_1,x_2,...x_n ~iid Unif([0,1]) .We have to find the probability P(x_1≤x_2≤...≤x_n ) = the probability that x_1 ≤ x_2 ... ≤ x_n .

Let's see that
P(x_1 ≤ x_2)  = ∫(x_2 = 0 to 1) f(x_2) (∫(x_1 = 0 to x_2) f(x_1)dx_1 ) dx_2  = 1/2! .

We can  find in similar way P(x_1 ≤  x_2 ≤ x_3) = 1/3!

So probability  to be found out = 1/n!

What is the probability that x = x_i one of the elements of x_1,x_2 ... x_n described before will be equals to the max(x_1,x_2,...x_n)?

Ok we have to find Σ ( for all value of x )P(x = x, x = max( ∀ x_i ∈ {x_1,x_2 ... x_n}  , x_i) )
=  P( x = max( ∀ x_i ∈ {x_1,x_2 ... x_n}  , x_i) ) 
=  1 - (F(x))^n

Where F(x) is the cdf of x.