A real number is picked at random from 01 Verify that event

A real number is picked at random from [0,1]. Verify that event A that the first
decimal digit of the number picked is 5 is independent of event B that the third decimal
digit is 1.

Solution

Real number X is picked from [0,1]

Consider any real number between them concisting of more than 3 digits as

even for 0 it is 0.000 and 0.1 as 0.1000 etc

First decimal number can be any one of 0 to 9 hence prob of A = 0.1

P(A) =prob of selecting 1 = 0.1

Third decimal number also is equally likely for all 0 to 9 and prob of B =0.1

Thus P(B/A) = P(B)

Hence B is independent of A