STAT 103.3Assignment 5Due 16 February 2022 4 March 2022 This assignment requires considerable numerical computation. You do not need to show detailed steps in the computation, and may use software tools to assist you. However, you should explain how you do the computation (in terms of words, or formulas) before givein the final answer. The graphs in Question 1 may be done by hand (neatly) or using software. Ensure that the axes are properly labelled. 1. Consider an urn containing 10 black and white balls of which three are black. Suppose that the balls are are numbered 1-10, with the black balls being number 1-3. (a) A sample of five balls is taken, one at a time, without replacement, such that all the balls in the urn at any time have equal probabilities of being selected. i. How many different sequences of balls can be obtained by this protocol? Are they all equally likely? ii. How many different samples of balls can be so obtained, where a sample is defined as a collection of numbered balls, disregarding the order in which they were drawn. Are they all equally likely? iii. If we define a result as the number of black and white balls drawn, disregarding the numbers of the balls, and the order in which they were drawn, how many possible differetn results are there? Are they equally likely.? iv. How many of these samples (i.e., noting ball numbers but ignoring order) will contain no black balls? How many will contain exactly one black ball? How many will contain two? Three? Four? Five? v. Draw the probability histogram and cumulative distribution function of the random variable X = {Number of black balls sampled }. (b) Repeat the question by assuming that the balls are sampled with replacement. Note that in this case the same numbered ball may appear more than once. If so, count each ball (and its colour) the number of times that it is drawn. 2. A resort has five cottages. Four families decide to book a cottage. The booking website randomly assigns each family a cottage, but a software glitch neglects to check whether a cottage has been already booked before assigning a family to it. (a) What is the probability that all the families are assigned to different cottages? (b) What is the proability that all the families are assigned to the same cottage? (c) Define the random variable X = {The number of families who have not been double booked with another family}. Calculate the probability mass function and cumulative distribution function of X. 3. Supposing that you are given a hand of three cards for a well-shuffled standard deck1 of 52. Assign numbers to each card as follows: Numbered cards are assigned the same number of points as their number, an ace is worth one point, and face cards (Jack, Queen, King) are worth 10 points each. (a) Let X represent the total number of points in your hand. Calculate the probability mass function of X. (b) Let Y represent the number of hearts in your hand. Calculate the probability mass function of Y . (c) Are X and Y independent random variables? (Hint: Conisder whether the suits of the cards make a difference in computing X.)

1Note to the uninitiated: A standard deck consists of 52 cards, consisting of four suits of 13 cards each. The suits are called clubs, diamonds, hearts, and spades. The 13 cards in a suit consist of an ace, nine cards numbered 2-10, a Jack, a King, and a Queen.