A Haunted Halloween game consists of all TAs flipping 8 fair coin in succession. In the event that all 8 coins on a TA's turn land on heads, legend has it that that TA will turn into a ghost on October 31. Let us define the following events in the resulting probabilistic space:

A: event that the first toss result in Tails.

B: event that the fourth toss results in Heads.

C: event that there are a total of 4 Heads.

D: event that the longest repeated sequence (sequence of repeated Heads or Tails) is at least length 2.

E: event that the longest repeated sequence (sequence of repeated Heads or Tails) is at most length 7.

Answer the following questions giving proper justification. (a) Are events A and B independent? (b) Are events A and C independent? (c) Are events B and C independent? (d) Are events A, B, and C mutually independent? (e) Are events D and E independent?