(a) We play a game where we can roll a single dice 100 times. Every time we roll two numbers in a row that sum to 7 we win $1. What are the expected winnings?

(b) A teacher has an empty book in her office with exactly 26 empty pages. She also has dictionary with twenty six words that start with the letter a, twenty six words that start with the letter b, etc. Since there are 26 letters, the dictionary has exactly two hundred and sixty two. Her goal is to randomly select a word from the dictionary and copy it to a randomly selected page of the book. However each page can have at most one word starting with each letter of the alphabet. If a page already contains a word starting with same letter as the word she selected, then that word is thrown away. Find the probability that the first page of the book contains a word starting with the letter a. Next find the expected number of words that will be in her new book.

(c) A box manufacturing company creates closed boxes of random lengths, widths, and heights. The length of the box is selected randomly from 2cm, 3cm, 4cm, or 5cm. The width of the box is determined by flipping a coin 5 times and counting the number of heads and then adding 1, (the result is in centimeters). The height of the box is either 5cm or 10cm (equally likely). Find the expected surface area and expected volume of this box. Now suppose the manufacturer decides that the length and width will be computed as above, but the height will be exactly half of the length. Find the new expected surface area and volume.

(d) Twenty people are standing in line at the bank. I randomly hand out either black or white hats to each person. Each person can only see their own hat and the hat of the person directly in front of them. What is the expected number of people who are standing behind someone wearing a different colored hat than their own?

(e) In a country with 20 cities, the government will build a road between any two pair of cities with probability 0.2. 4 What is the expected number of roads coming out of any given city? What is the expected number of roads in total? What is the probability that there is a road from city A to B to C? Suppose that before construction, the government realizes that they only have enough money to build exactly 80 roads, and they will randomly select which cities to connect. What is the expected number of roads coming out of any given city? What is the probability that there is a road from city A to B to C?