1. * (Distribution of Maximum). Suppose n numbers are drawn at random from f1; 2; ··· ;N g. What is the probability that the largest number drawn is a specified number k if sampling is (a) with replacement; (b) without replacement?

2. * (Poisson Approximation). One hundred people will each toss a fair coin 200 times. Approximate the probability that at least 10 of the 100 people would have obtained exactly 100 heads and 100 tails.

3. * (A Design Problem). You are allowed to choose a number n and then toss a fair coin ntimes. You will get a prize if you can get either seven or nine heads. What is your best choice of the numbern?

4. * (A Novel Way to Give a Test). A student takes a five-answer multiple-choice oral test. His grade is determined by the number of questions re- quired in order for him to get five correct answers. A grade of A is given if he requires only five questions; a grade of B is given if he requires six or seven ques- tions; a grade of C is given if he requires eight or nine questions; and he fails otherwise.

Suppose the student guesses independently at random on each question. What is his most likely grade?