4. Counting Categories In each part below, write the probabilities as math expressions. You don't have to find any numerical values. But please make sure there are no infinite sums in your answers. For distributions, you can just provide names and parameters if they are applicable. a) 18 dice are rolled. Let X be the number of times the face with 1 spot appears. Let Y be the number of times a multiple of 3 appears. What is the joint distribution of X and Y? [Be careful about the possible values of the pair (X, Y).] b) 18 dice are rolled. Find the chance that two of the faces appear 5 times each, another two faces appear 3 times each, and the remaining two faces appear 1 time each. c) Repeat Part b in the case when the number of dice rolled isn't 18 but is instead a random number that has the Poisson distribution. d) As in Part c, roll a random number of dice, where the random number has a Poisson distribution. Don't touch any die that shows the face with six spots. If there are any other dice (that is, if there are dice that don't show six spots), roll those dice one more time. Then stop. Let S be the total number of dice that show the face with six spots when you stop. Find the distribution of S