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We roll two dice. Assume all 36 possibilities are equally likely. Let X1 and X2 be...
3. Two fair, four-sided dice are rolled. Let X1, X2 be the outcomes of the first...
3. Two fair, four-sided dice are rolled. Let X1, X2 be the outcomes of the first and second die, respectively. (a) Find the conditional distribution of X2 given that Xi + X2 = 4. (b) Find the conditional distribution of X2 given that Xi + X2-5.
What is the most likely outcome when we throw two fair dice, i.e., what is the most likely sum th...
What is the most likely outcome when we throw two fair dice,
i.e., what is the most likely sum that the two dice would add to?
Why? This problem can be solved by first principles. The probability
P(E) for an event E is the ratio |E|/|S|, where |E| is the
cardinality of the event space and |S| is the cardinality of the
sample space. For example, when we throw a fair die, the event
space is S = {1,2,3,4,5,6} and...
Probability and Random Processes for Engineers You roll a fair die twice: all 36 outcomes are...
Probability and Random Processes for
Engineers
You roll a fair die twice: all 36 outcomes are equally likely. Let A be the event that the first roll is 1, 2, or 3. Let B be the event that the second roll is 6. Finally, let C be the event that the sum of the rolls is even. (a) Show that any two of A, B, and C are independent (b) Are A, B, and C independent? Derive your answer two...
(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) Fi...
(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) Fi...
(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...
For the two six-sided dice case: Write out the six-by-six matrix showing all possible (36) combinations...
For the two six-sided dice case: Write out the six-by-six matrix showing all possible (36) combinations of outcomes. Draw a histogram of the probability of outcomes for the dice totals. Explain the shape of the histogram. Draw a Venn diagram for the 36 dice roll combinations. Define a set "A" as all the combinations that total seven; define set "B" as all the combinations that have one die roll (either die 1 or 2) equal to 2. Indicate the sets...
The Dice game of "Pig" can be played with the following rules. 1. Roll two six-sided dice. Add the face values...
The Dice game of "Pig" can be played with the following rules. 1. Roll two six-sided dice. Add the face values together. 2. Choose whether to roll the dice again or pass the dice to your opponent. 3. If you pass, then you get to bank any points earned on your turn. Those points become permanent. If you roll again, then add your result to your previous score, but you run the risk of losing all points earned since your...
Roll two fair four-sided dice. Let X and Y be the die scores from the 1st...
Roll two fair four-sided dice. Let X and Y be the die scores from the 1st die and the 2nd die, respectively, and define a random variable Z = X − Y (a) Find the pmf of Z. (b) Draw the histogram of the pmf of Z. (c) Find P{Z < 0}. (d) Are the events {Z < 0} and {Z is odd} independent? Why?
Q1 (100) Suppose you roll two twenty-five-sided dice. Let X1, X2 the outcomes of the rolls...
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.
3) We roll 2 fair dice. a) Find the probabilities of getting each possible sum (i.e....
3) We roll 2 fair dice. a) Find the probabilities of getting each possible sum (i.e. find Pr(2), Pr(3), . Pr(12) ) b) Find the probability of getting a sum of 3 or 4 (i.e.find Pr(3 or 4)) c) Find the probability we roll doubles (both dice show the same value). d) Find the probability that we roll a sum of 8 or doubles (both dice show the same value). e) Is it more likely that we get a sum...
3. Two fair, four-sided dice are rolled. Let X1, X2 be the outcomes of the first and second die, respectively. (a) Find the conditional distribution of X2 given that Xi + X2 = 4. (b) Find the conditional distribution of X2 given that Xi + X2-5.
What is the most likely outcome when we throw two fair dice,
i.e., what is the most likely sum that the two dice would add to?
Why? This problem can be solved by first principles. The probability
P(E) for an event E is the ratio |E|/|S|, where |E| is the
cardinality of the event space and |S| is the cardinality of the
sample space. For example, when we throw a fair die, the event
space is S = {1,2,3,4,5,6} and...
Probability and Random Processes for
Engineers
You roll a fair die twice: all 36 outcomes are equally likely. Let A be the event that the first roll is 1, 2, or 3. Let B be the event that the second roll is 6. Finally, let C be the event that the sum of the rolls is even. (a) Show that any two of A, B, and C are independent (b) Are A, B, and C independent? Derive your answer two...
(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...
The Dice game of "Pig" can be played with the following rules. 1. Roll two six-sided dice. Add the face values together. 2. Choose whether to roll the dice again or pass the dice to your opponent. 3. If you pass, then you get to bank any points earned on your turn. Those points become permanent. If you roll again, then add your result to your previous score, but you run the risk of losing all points earned since your...
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.
3) We roll 2 fair dice. a) Find the probabilities of getting each possible sum (i.e. find Pr(2), Pr(3), . Pr(12) ) b) Find the probability of getting a sum of 3 or 4 (i.e.find Pr(3 or 4)) c) Find the probability we roll doubles (both dice show the same value). d) Find the probability that we roll a sum of 8 or doubles (both dice show the same value). e) Is it more likely that we get a sum...
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