1) i) The fractional parts of 100 numbers are distributed uniformly between 0 and '|.The numbers are first rounded to the nearest integer and then added. Using the CLT. nd an approximation for the probability that the error in the sum due to rounding lies between 0.5 and 0.5. Hint: First argue that the individual errors are uniformly distributed and find the mean and variance ofthe uniform distribution. ii) A man buys a new die and throws it 600 times. He does not yet know if the die is fair! a) [2 marks] Show that the probability that he obtains between 90 and 100 'sixes' if the die is fair is 0.3968 approximately. b) [3 marks] We see from (a) that P(90 S X S 100) =3 0.3968 where X denotes the number of 'sixes' from 600 throws. If the die is fair we can expect about '|00 'sixes' from 600 throws. Between what two limits symmetrically placed about '| 00 would the number sixes obtained lie with probability 0.95 ifthe die is fair? That is, nd N such that PUOO NS X S 100 + N) = 0.95. You may use the continuity correction of 0.5 on each ofthe limits inside the probability statement. c) [2 marks] What might he conclude if he obtained 120 sixes