代写SYS 6005 - Homework Assignment 1代做Python编程

SYS 6005 - Homework Assignment 1

Due: 15 September 2024 @ 11:59 PM

Problems to Turn In: (Total 100 points)

Instructions: Solve each of the following exercises. Give quantitative answers where required, and always explain your reasoning.

Problem 1: (25 points)

Alice and Bob each choose at random a real number between zero and three. We assume a uniform. probability law under which the probability of an event is proportional to its area. Consider the following events:

A: The magnitude of the difference of the two numbers is greater than 1.

B: At least one of the numbers is greater than 1 2/1.

C: The numbers are equal.

D: Alice’s number is greater than 2.

Find the probabilities P(B ∩ D), P(A ∩ Bc ), P(C|B ∩ D), P(D|A), P(A|Cc).

Problem 2: (25 points)

Alice and Bob have 7 coins, each with the probability of a head equal to p = .7. Bob tosses 3 coins, while Alice tosses the remaining 4 coins. Assuming that all tosses are independent, compute the probability that Alice gets more heads than Bob.

Problem 3: (25 points)

Consider an experiment in which a fair six-sided die (with faces labeled 1, 2, 3, 4, 5, 6) is rolled once to determine how many times a four-sided fair die (with faces labeled 1, . . . , 4) is rolled. The die rolls are independent. Let Y be the result of the six-sided die roll, and let X be the total number of times the four-sided die results in 4.

(a) (5 points) Calculate the conditional probability that X = 2 given that the four-sided die roll resulted in 2.

(b) (6 points) Calculate P({X = 2}).

(c) (6 points) Given that the experiment resulted in X = 2, what is the probability that the result of the four-sided die roll was 3?

Consider now the same experiment repeated several times, where the die rolls in different experiments are independent. Let Yi be the result of the six-sided die roll in the i-th experiment, and let Xi be the total number of times the four-sided die results in 4 in the i-th experiment.

(d) (4 points) Are events {X1 = 2} and {X2 = 2} independent?

(e) (4 points) Given {Y1 = Y2}, are events {X1 = 2} and {X2 = 2} independent?

Problem 4: (25 points)

We are given three coins. The first coin is a fair coin painted blue on the head side and white on the tail side. The other two coins are biased so that the probability of a head is p. They are painted blue on the tail side and red on the head side. Two of the three coins are to be selected at random and tossed. Describe the outcomes in the sample space. What is the probability that the sides that land face up are of the same color? Your answer can be in terms of p.