Instructions: This homework addresses the material covered in Lecture 1, on §1.1–§1.3 in the text . Recommended supplemental problems from text. These problems are not to be handed in with your assignment. You are encouraged to work with your classmates on the supplemental problems. • §1.2. Probabilistic models: Problems 6, 7, 8, 9, 10. • §1.3. Conditional probability: Problems 15, 16. Required problems for homework: 1. We roll a four-sided die once and then we roll it as many times as is necessary to obtain a diﬀerent face than the one obtained in the ﬁrst roll. Let the outcome of the experiment be (r1,r2) where r1 and r2 are the results of the ﬁrst and the last rolls, respectively. Assume that all possible outcomes have equal probability. Find the probability that: (a) r1 is even (b) Both r1 and r2 are even. (c) r1 + r2 < 5. 2. A magical four-sided die is rolled twice. Let S be the sum of the results of the two rolls. We are told that the probability that S = k is proportional to k, for k = 2,3,··· ,8, and that all possible ways that a given sum k can arise are equally likely. Construct an appropriate probabilistic model and ﬁnd the probability of getting doubles. 3. Alice and Bob each choose at random a number between zero aand two. We assume a uniform probability law under which the probability of an event is proportional to its area. Consider the following events: (a) A: The magnitude of the diﬀerence of the two numbers is greater than 1/3. (b) B: At least one of the numbers is greater than 1/3. (c) C: The two numbers are equal. (d) D: Alice’s number is greater than 1/3. Find the probabilities P(A),P(B),P(A∩B),P(C),P(D),P(A∩D). 4. Consider a game where two fair (unbiased) coins are tossed, repeatedly as necessary as described below. Each toss results in heads (two heads), tails (two tails), or odds (one of each). A bet can be made that the spinner (coin tosser) will obtain three heads before a single tails and before ﬁve consecutive odds. Notice that at most 15 tosses are required to resolve this bet. (If it is unresolved after 14 tosses, then the 14 tosses must consist of oooohoooohoooo, and the next toss is decisive.) Write a computer program to simulate this game, and run your program 100,000 times keeping track of the fraction of n plays that result in a win. Plot this fraction of winning plays versus n, for n = 100,200,…,99900,100000. That is, every 100 plays you should compute the cumulative fraction of plays that have been wins, and plot that fraction vs. n. What is your ﬁnal estimate of the probability of winning this bet. Make sure your plot has labeled axes and a title.