If you just had to predict the first digit of x100 what

Question

Let Xi be the last digit of D2i where Di is a random digit between 0 and 9. For instance, if D_i = 7 then D²_i = 49 and X_i = 9, Let X_n = (X₁ +  ..  X_n)/n be the average of a large number n of such last digits, obtained from independent random digits D₁,  . ..  D_n.

a) Predict the value of X_n for large n.

b) Find a number ? such that for n = 10,000 the chance that your predication is off by more than ? is about 1 in 200.

c) Find approximately the least value of n such that your prediction of X_n is correct to within 0.01 with probability at least 0.99.

d) Which can be predicted more accurately for large n: the value of X_n, or the value of D_n = (D₁ + . .. +D_n)/n?

e) If you just had to predict the first digit of X₁₀₀, what digit should you choose to maximize your chance of being correct, and what is that chance?