![Problem 1. (12 points) A fair 6-sided die is rolled three times. Which is more likely: a sum of 11 or a sum of 12? Answer the question by calculating the probabilities for both. [hint 1] There are multiple ways to solve this problem. You may list all the favorable permutations to get the sum. However, this might be tedious and more error-prone. An easier way is to list only the favorable combinations (i.e., 3 numbers regardless of their order), and then find out the corresponding numbers of permutations for each combination. Thint 2] In a combination corresponding permutations (2-5-5, 5-2-5, and 5-5-2) tian in which two numbers are the same (e.g., 2, 5, 5), there are 3](http://img.homeworklib.com/questions/3e959b90-70fa-11ea-b52f-c1a0570f9fee.png?x-oss-process=image/resize,w_560)
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Problem 1. (12 points) A fair 6-sided die is rolled three times. Which is more likely:...
A fair 6-sided die is rolled three times. Which is more likely: a sum of 11...
A fair 6-sided die is rolled three times. Which is more likely: a sum of 11 or a sum of 12? Answer the question by calculating the probabilities for both. Thint 1] There are multiple ways to solve this problem. You may list all the favorable permutations to get the sum. However, this might be tedious and more error-prone. An easier way is to list only the favorable combinations (i.e., 3 numbers regardless of their order), and then find out...
7. (3 points) Given a fair 6-sided die. Each time the die is rolled, the probabilities...
7. (3 points) Given a fair 6-sided die. Each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 6 are all equal. 1) If it is rolled once and let A be the event of rolling a number larger than 3 and B be the event of rolling an odd number. What is P(AV B)? 2) If it is rolled three times, what is the probability that the same number shows up in...
Problem 2. A tetrahedron four-sided die) is rolled turice and the sum X of the results...
Problem 2. A tetrahedron four-sided die) is rolled turice and the sum X of the results of the two rolls is recorded. We know that the chance that X-k is proportional to k. (a) What is the probability model for X, i.e., what values can X take, and what are the corresponding probabilities? (b) Compute the chance that the sum of the two rolls wieceed but will not be more than 6. (c) Compute the expected value of X
A six sided die is rolled three times independently. How many different ways can you get...
A six sided die is rolled three times independently. How many different ways can you get a sum of 11? sum of 12?
A fair 6-sided die rolled 5 times. what is the probability that at least one of...
A fair 6-sided die rolled 5 times. what is the probability that at least one of the rolls is 2
Problem #5: A fair 8-sided die is rolled 101 times. Consider the event A = {the...
Problem #5: A fair 8-sided die is rolled 101 times. Consider the event A = {the face 2 comes up at most 2 times) (a) Find the normal approximation for P(A) without the continuity correction. (b) Find the normal approximation for P(A) with the continuity correction. (c) Find the Poisson approximation for P(A).
6. A fair six sided die is rolled three times. Find the probability that () all...
6. A fair six sided die is rolled three times. Find the probability that () all three rolls are either 5 or 6 (6) all three rolls are even (c) no rolls are 5 (d) at least one roll is 5 (e) the first roll is 3, the second roll is 5 and the third roll is even
Problem 3. (10 points) We roll two fair 6-sided dice. (1) What is the probability that...
Problem 3. (10 points) We roll two fair 6-sided dice. (1) What is the probability that at least one die roll is 6? (2) Given that two two dice land on different numbers, what is the conditional probability that at least one die roll is a 6? Thint] You may use the graphical approach (Lecture 5 slide 11-12) to help you solve the problem.
Problem #3: 5 fair 12-sided dice are rolled. (a) [3 marks] Find the conditional probability that...
Problem #3: 5 fair 12-sided dice are rolled. (a) [3 marks] Find the conditional probability that at least one die lands on 3 given that all 5 dice land on different numbers. 6) [2 marks] True or False: If X is the maximum of the 5 numbers from one roll, and Y is the minimum of the 5 numbers from one roll, then X and Y are independent random variables.
Problem 1. Suppose that a fair six-sided die is rolled n times. Let N be the number of 1's rolled...
I know Pk~1/k^5/2 just need the
work
Problem 1. Suppose that a fair six-sided die is rolled n times. Let N be the number of 1's rolled, N2 be the number of 2's rolled, etc, so that NN2+Ns-n Since the dice rolls are independent then the random vector < N,, ,Ne > has a multinomial distribution, which you could look up in any probability textbook or on the web. If n 6k is a multiple of 6, let Pa be...
A fair 6-sided die is rolled three times. Which is more likely: a sum of 11 or a sum of 12? Answer the question by calculating the probabilities for both. Thint 1] There are multiple ways to solve this problem. You may list all the favorable permutations to get the sum. However, this might be tedious and more error-prone. An easier way is to list only the favorable combinations (i.e., 3 numbers regardless of their order), and then find out...
7. (3 points) Given a fair 6-sided die. Each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 6 are all equal. 1) If it is rolled once and let A be the event of rolling a number larger than 3 and B be the event of rolling an odd number. What is P(AV B)? 2) If it is rolled three times, what is the probability that the same number shows up in...
Problem 2. A tetrahedron four-sided die) is rolled turice and the sum X of the results of the two rolls is recorded. We know that the chance that X-k is proportional to k. (a) What is the probability model for X, i.e., what values can X take, and what are the corresponding probabilities? (b) Compute the chance that the sum of the two rolls wieceed but will not be more than 6. (c) Compute the expected value of X
Problem #5: A fair 8-sided die is rolled 101 times. Consider the event A = {the face 2 comes up at most 2 times) (a) Find the normal approximation for P(A) without the continuity correction. (b) Find the normal approximation for P(A) with the continuity correction. (c) Find the Poisson approximation for P(A).
6. A fair six sided die is rolled three times. Find the probability that () all three rolls are either 5 or 6 (6) all three rolls are even (c) no rolls are 5 (d) at least one roll is 5 (e) the first roll is 3, the second roll is 5 and the third roll is even
Problem 3. (10 points) We roll two fair 6-sided dice. (1) What is the probability that at least one die roll is 6? (2) Given that two two dice land on different numbers, what is the conditional probability that at least one die roll is a 6? Thint] You may use the graphical approach (Lecture 5 slide 11-12) to help you solve the problem.
Problem #3: 5 fair 12-sided dice are rolled. (a) [3 marks] Find the conditional probability that at least one die lands on 3 given that all 5 dice land on different numbers. 6) [2 marks] True or False: If X is the maximum of the 5 numbers from one roll, and Y is the minimum of the 5 numbers from one roll, then X and Y are independent random variables.
I know Pk~1/k^5/2 just need the
work
Problem 1. Suppose that a fair six-sided die is rolled n times. Let N be the number of 1's rolled, N2 be the number of 2's rolled, etc, so that NN2+Ns-n Since the dice rolls are independent then the random vector < N,, ,Ne > has a multinomial distribution, which you could look up in any probability textbook or on the web. If n 6k is a multiple of 6, let Pa be...
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