1 [20 marks total] For COMP2610 and COMP6261 Students An urn contains balls, of which are black and = are white. Frank draws a ball at random from the urn and replaces it, times. (a) What is the probability distribution of the number of times a Black ball is drawn, ? What is the probability distribution of the number of times a white ball is drawn, ? Give answers in terms of , , and . [4 marks] (b) What is the expectation of ? What is the variance of ? What is the standard deviation of ? Give answers in terms of , , and . [4 marks] (c) Give numerical answers of the expectation, variance, and standard deviation of for the cases = 10 and = 100, when = 3 and = 10. [4 marks] (d) We define the fraction /. Frank draws times from the urn, same as in (a) and (b), obtaining black balls, and computes the quantity = ( )2 (1 ) . What is the expectation of ? [4 marks] (e) In the case = 5 and = 1/5, what is the probability that < 1