Tossing a fair and unfair die. Suppose you have a 8-sided die, with sides numbered 1, 2, 3, 4, 5, 6, and 7, 8. (a) To write a program (in any programming language) to simulate the tossing of a 8- sided fair die (i.e., all sides are equally likely), for t =50, 100, 1000, 2000 and 3000 tosses. Based on the simulation, what is the estimated probability of obtaining an odd number? Please provide screenshots of the estimated probability outputs that you get when you run your code.

(b) Suppose X is a random variable denoting the outcome of a die toss. Based on the mathematical analysis, what is the probability that X has odd value? Please provide a numerical value of the probability (by solving mathematically by hand). Show all your work for finding the probability.

(c) Refer back to part (a). Does it agree with the theoretical result in (b)? Please provide a yes or no answer for each value of t used in part (a). Explain your observation.

(d) Repeat parts (a), (b), and (c) with a 8-sided die that has the following properties: The probability of the die outcome being 1 is equal to the probability of the die outcome being 2. The probabilities of the die outcome being a 3, 4, 5, 6, 7 or 8 are all equal. The probability of the die outcome being a 1 is twice the probability of the die outcome being 8.

Answer D alone