Assume E is the event that a randomly generated bit string of length 4 starts with a 1 and F is the event that this bit string consists of an even number of 1's. Are E and F independent if the 16 bit strings of length 4 are equally likely? 

Ans: Number of 4 bit strings that starts with 1 is 8, thus P(E) = .5  

Number of 4-bit string comprising even number of 1's is also 8 [C (4, 0) + C(4, 2) + C(4, 4)]. Hence P(F) = .5  

Here now Number of 4-bit string that begins with 1 and consists of even number of 1's is 4 [1 + C(3, 0)]. So P (E∩F) = .25.

Obviously P (E∩F) = P(E)*P(F). So E and F are independent.