Question: Let Xi be the last digit of D2i where Di is a random digit between 0 and 9. For instance, if Di = 7 then D2i = 49 and Xi = 9, Let Xn = (X1 + .. Xn)/n be the average of a large number n of such last digits, obtained from independent random digits D1, . .. Dn.
a) Predict the value of Xn 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 Xn 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 Xn, or the value of Dn = (D1 + . .. +Dn)/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?