/ 1 80% + Help Problem 1: Short Answers about Probability (a) Alice declares that the probability for rain tomorrow is 30%. To be consistent, what should she declare to be the probability for not raining? (b) A novelty-toy manufacturer sells a loaded die that is described by the following prob- abilities: x: 1 2 3 4 5 6 4 p(x): 12 12 (1) What is the probability that the result of rolling the die will be 3 or less? (c) What is the probability that the result of rolling the die from part (b) will be an odd number? (d) Bob flips a fair coin four times in a row. What is the probability that he will get four heads? (e) Alice and Bob both order soup from a restaurant that sells five kinds of soup. Assuming that both of them make their choice at random, what is the probability that they will pick the same kind of soup? Problem 2: Permutations and Combinations (a) In the movie Dr. Strangelove: Or, How I Learned to Stop Worrying and Love the Bomb (1964), B-52 bomber planes are equipped with a device that prevents them from receiving radio transmissions, unless that transmission begins with the proper three-letter code prefix. How many possible prefixes are there? (b) A bakery sells six different types of cookie. The baker wishes to arrange one cookie of each type in a row across the counter. How many different ways are there to do so? (c) Alice visits the bakery from part (b) and wants to buy three different kinds of cookie. How many different options does she have? Problem 3: Weighted Averages (a) Consider again the loaded die from Problem 1(b). What is the expected value (r)? (b) What is the Shannon entropy S of that probability distribution?