Assume that a big tub has 1000 soft toys. Suppose that the soft toys are numbered from 1 to 1000.

(a) If we draw 1000 soft toys, with replacement, what is the approximate probabiity to observe the soft toy numbered 1 less than 3 times? [3 marks]

(b) Find the expected number of times the soft toy numbered 1 is observed, when the 1000 picks are with replacement. [4 marks]

(c) If we draw 1000 soft toys, without replacement, what is the probability that the soft toy numbered 2 is picked before the one numbered 51? [3 marks]

(d) If we draw only 50 soft toys out of the 1000, without replacement, find the expected number of times the soft toy numbered 1 is observed. [3 marks]

(e) Still consider the case when we draw only 50 soft toys out of the 1000, without replacement. Denote by S the total sum of the numbers observed in the 50 trials. Find the mean of S. Compare to the case where there is replacement. [5 marks]