3. Two fair, four-sided dice are rolled. Let X1, X2 be the outcomes of the first...
(4) Consider rolling three dice. Let X1, X2, and X3 the values which appear on the three dice, respectively. Let Y be the maximum out of all three dice. (a) Find the conditional PMF py)xi (yr) (b) Find the probability that the maximum of all three dice is 4 given that the first die is a 3. (c) Find the probability that the maximum of all three dice is 3 given that the first die is a 3 (4) Consider...
(4) Consider rolling three dice. Let X1, X2, and X3 the values which appear on the three dice, respectively. Let Y be the maximum out of all three dice. (a) Find the conditional PMF py)xi (yr) (b) Find the probability that the maximum of all three dice is 4 given that the first die is a 3. (c) Find the probability that the maximum of all three dice is 3 given that the first die is a 3 (4) Consider...
Two identical fair 6-sided dice are rolled simultaneously. Each die that shows a number less than or equal to 4 is rolled once again. Let X be the number of dice that show a number less than or equal to 4 on the first roll, and let Y be the total number of dice that show a number greater than 4 at the end. (a) Find the joint PMF of X and Y . (Show your final answer in a...
Q1 (100) Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls of these two fair dice which can be viewed as a random sample of size 2 from a uniform distribution on integers. a) What is population from which these random samples are drawn? Find the mean (u) and variance of this population (62)? Show your calculations and results.
Three six-sided fair dice are rolled. The six sides are numbered 1,2,3,4,5,6. Let A be the event that the first die shows an even number, let B be the event that the second die shows an even number, and let C be the event that the third die shows an even number. Express each of the following events in terms of the named events described above: 1) the event that all three dice show even numbers 2) the event that...
Conditional Probability Two fair dice are rolled: (a) Express the sample space S in set builder notation and the probability P "At least one of the dice rolls a four." Write all possible outcomes of A (b) Consider the event A (c) What is the probability that at least one die rolls a four? (d) What is the conditional probability that the first die rolls a four given that the sum of the dice is six? (e) What is the...
A die is rolled twice. Let X1 and x2 be the outcomes, and let s= X1 + x2 be the sum of these outcomes. 1. What is the expected value of x1 and x2? 2. What is the standard deviation of x1 and x2? 3. What is the expected value and standard deviation of sA die is rolled twice. Let X1 and x2 be the outcomes, and let s= X1 + x2 be the sum of these outcomes. 1. What...
In this experiment, both a fair four-sided die and a fair six-sided die are rolled (these dice both have the numbers most people would expect on them). Let Z be a random variable that represents the absolute value of their difference. For instance, if a 4 and a 1 are rolled, the corresponding value of Z is 3. (a) What is the pmf of Z? (b) Draw a graph of the cdf of Z
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.
A) Suppose I roll two fair six-sided dice. What is the probability that I rolled a total of 5? B) Suppose I roll two fair six-sided die and I announce that the sum of the two die is 6 or less. What is the probability that I rolled a total of 5?