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7. Consider the following gamd. We start with an urn with 1 red ball and 1...
7. Consider the following gamel. We start with an urn with 1 red ball and 1...
7. Consider the following gamel. We start with an urn with 1 red ball and 1 blue ball. Each round, we reach in and grab a ball at random, then return that ball plus one more ball of the same color. Repeat this process until there are n balls in the bin. Show that the number of red balls is equally likely to be any number between 1 and n - 1.
Consider the following game We start with an urn with 1 red ball and 1 blue...
Consider the following game We start with an urn with 1 red ball and 1 blue ball. Each round, we reach in and grab a ball at random, then return that ball plus one more ball of the same color Repeat this process until there are n balls in the bin. Show that the number of red balls is equally likely to be any number between 1 and n -1.
Consider the following game. We start with an urn with 1 red ball and 1 blue...
Consider the following game. We start with an urn with 1 red ball and 1 blue ball. Each round, we reach in and grab a ball at random, then return that ball plus one more ball of the same color. Repeat this process until there are n balls in the bin. Show that the number of red balls is equally likely to be any number between 1 and n−1.
Consider the following gam. We start with an urn with 1 red ball and 1 blue...
Consider the following gam. We start with an urn with 1 red ball and 1 blue ball. Each round, we reach in and grab a ball at random, then return that ball plus one more ball of the same color. Repeat this process until there are n balls in the bin. Show that the number of red balls is equally likely to be any number between 1 andn-1
3. * One urn contains one black ball and one red ball. A second urn contains...
3. * One urn contains one black ball and one red ball. A second urn contains one white ball and one red ball. One ball is selected at random from each urn (a) Exhibit the sample space ? for this experiment. (b) Show the subset of ? that defines the event A that both balls will be of the same color. (c) Assume each elementary outcome in 2 is equally likely to occur. What is the probability that both balls...
2. Consider an urn that contains red and green balls. At time 0 there are k balls with at least one ball of each color. At time n we draw out a ball chosen at random.We return it to the urn and add o...
2. Consider an urn that contains red and green balls. At time 0 there are k balls with at least one ball of each color. At time n we draw out a ball chosen at random.We return it to the urn and add one more of the color chosen. Let X be the fraction of red balls at time n. Show that Xn is a martingale with respect to the filtration (X0,Xi, ,Xn). At time n there are nk balls,...
5. Polya's Urn +R • Begin with an urn containing W white balls and R red...
5. Polya's Urn +R • Begin with an urn containing W white balls and R red balls. Hence, N =W total balls. • Draw a random ball from the urn, check its color, and return it to the urn with another ball of the same color. • Now there are N + 1 balls in the urn. Draw 1 at random, check its color, and return it with another ball of the same color. . Now there are N +...
Urn 1 contains 3 red and 6 blue balls, and urn 2 contains 4 red and...
Urn 1 contains 3 red and 6 blue balls, and urn 2 contains 4 red and 3 blue balls. The urns are equally likely to be chosen. a) If a blue ball is drawn, what is the probability that it came from urn 1? b) If a red ball is drawn, what is the probability that it came from urn 2?
An urn contains 2 balls that are either red or blue. At each step a ball...
An urn contains 2 balls that are either red or blue. At each step a ball is randomly drawn and replaced with a new ball, having the same color w.p. 4/5, or different color w.p. 1/5. Find the probability that the 5th ball drawn is red, if you start with 2 red balls in the urn. Please explain step by step how the transition probability matrix is formed.
In an urn, there are 3 blue balls and 2 red balls. A ball is selected...
In an urn, there are 3 blue balls and 2 red balls. A ball is selected at random without replacement. Regardless of what color the first ball was, a blue ball is added to the urn (so there are a total of 5 balls in the urn now). Then a second ball is selected from the urn. a) What is the chance that the second ball selected from the urn is blue? b) Given that the second ball is blue,...
7. Consider the following gamel. We start with an urn with 1 red ball and 1 blue ball. Each round, we reach in and grab a ball at random, then return that ball plus one more ball of the same color. Repeat this process until there are n balls in the bin. Show that the number of red balls is equally likely to be any number between 1 and n - 1.
Consider the following game We start with an urn with 1 red ball and 1 blue ball. Each round, we reach in and grab a ball at random, then return that ball plus one more ball of the same color Repeat this process until there are n balls in the bin. Show that the number of red balls is equally likely to be any number between 1 and n -1.
Consider the following gam. We start with an urn with 1 red ball and 1 blue ball. Each round, we reach in and grab a ball at random, then return that ball plus one more ball of the same color. Repeat this process until there are n balls in the bin. Show that the number of red balls is equally likely to be any number between 1 andn-1
3. * One urn contains one black ball and one red ball. A second urn contains one white ball and one red ball. One ball is selected at random from each urn (a) Exhibit the sample space ? for this experiment. (b) Show the subset of ? that defines the event A that both balls will be of the same color. (c) Assume each elementary outcome in 2 is equally likely to occur. What is the probability that both balls...
2. Consider an urn that contains red and green balls. At time 0 there are k balls with at least one ball of each color. At time n we draw out a ball chosen at random.We return it to the urn and add one more of the color chosen. Let X be the fraction of red balls at time n. Show that Xn is a martingale with respect to the filtration (X0,Xi, ,Xn). At time n there are nk balls,...
5. Polya's Urn +R • Begin with an urn containing W white balls and R red balls. Hence, N =W total balls. • Draw a random ball from the urn, check its color, and return it to the urn with another ball of the same color. • Now there are N + 1 balls in the urn. Draw 1 at random, check its color, and return it with another ball of the same color. . Now there are N +...
In an urn, there are 3 blue balls and 2 red balls. A ball is selected at random without replacement. Regardless of what color the first ball was, a blue ball is added to the urn (so there are a total of 5 balls in the urn now). Then a second ball is selected from the urn. a) What is the chance that the second ball selected from the urn is blue? b) Given that the second ball is blue,...
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