Suppose we have two urns (a left urn and a right urn). The left urn contains N black balls and the right urn contains N red balls.
Suppose we have two urns (a left urn and a right urn). The left urn contains N black balls and the right urn contains N red balls.
Every time step you take one ball (chosen randomly) from each urn, swap the balls, and place them back in the urns. Let Xm be the number of black balls in the left urn after m time steps.
Find the Markov chain model and find the unique stationary distribution when N=5
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Suppose we have two urns (a left urn and a right urn). The left urn contains N black balls and the right urn contains N red balls.
Example. 2 urns. Red urn contains 3 red balls, 2 white balls. White urn contains 1...
Example. 2 urns. Red urn contains 3 red balls, 2 white balls. White urn contains 1 red ball, 4 white balls. Pick an urn randomly. Randomly select a ball from that urn. Without replacing the 1st ball, select a ball from the urn whose color matches the first ball. Q. Make a tree diagram, complete with probabilities describing this situation. Q. Find the probability that the first ball is white. Q. Find the probability that the second ball is white....
Urn one contains two red, one black balls, urn two contains one red, three black balls,...
Urn one contains two red, one black balls, urn two contains one red, three black balls, and urn three contains one red, one black balls. A student chooses urn one or urn two at random, and selects one ball from the chosen urn at random and transfers it into urn three. Then he draws a ball from urn three. Given that the ball he draws is red, what is the probability that the transferred ball is red?
2, Five white balls and five black balls are distributed in two urns in such a...
2, Five white balls and five black balls are distributed in two urns in such a way that each urn contains five balls. At each step we draw one ball from each urn and exchange them. Let Xn be the number of white balls in the left urn at time n. Compute the transition probability for X,
Five white balls and two red balls are distributed between two urns A and B in...
Five white balls and two red balls are distributed between two urns A and B in such a way that urn A contains 2 white balls and 2 red balls and urn B contains 3 white balls. At each step, we draw one ball from each urn and place the ball drawn from urn A into urn B and conversely with the ball from urn B. Find the long run proportion of time that urn A will contain 4 white...
Consider 3 urns. The urn A contains 5 white balls and 10 red balls, the urn...
Consider 3 urns. The urn A contains 5 white balls and 10 red balls, the urn B contains 9 white balls and 6 red balls and the urn C contains 4 white balls and 9 red balls. A ball is selected from ballot box 1 and placed in ballot box 2, then one ball is taken from ballot box 2 and placed in ballot box 3. Finally, a ball is taken from ballot box 3. What is the probability that...
L. An un contains n red balls and n black balls. Balls are drawn sequentially from...
L. An un contains n red balls and n black balls. Balls are drawn sequentially from the urn one at a time withont replacement. Let the first black ball is chosen. Find EX X denote the number of red balls removed befor
8) For Polya's urn model with r red balls, b black balls and parameters cE N...
8) For Polya's urn model with r red balls, b black balls and parameters cE N (the number of extra balls we insert every time) and n EN,n 2 3 (the number of times we perform the a) Show that the probability that the k-th chosen ball (k e (1,...,n]) is red is equal to b) Given that the second chosen ball was red compute the probability that the first one experiment), do the following: was red as well.
4. Urn A contains 3 red and 3 black balls, whereas urn B contains 4 red...
4. Urn A contains 3 red and 3 black balls, whereas urn B contains 4 red and 6 black balls. If a ball is randomly selected from each urn, what is the probability that the balls will be the same color?
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?
Consider 2 urns. Urn 1 has 2 White Balls and 1 Black Ball. Urn 2 has...
Consider 2 urns. Urn 1 has 2 White Balls and 1 Black Ball. Urn 2 has 1 White Ball and 2 Black Balls. Suppose that one ball is randomly drawn from Urn 1 and put into Urn 2. Then balls are selected one at a time without replacement from Urn 2 until a White Ball is obtained. Let Y be the number of balls drawn from Urn 2 until a white ball is drawn. Find the pdf of Y and...
Example. 2 urns. Red urn contains 3 red balls, 2 white balls. White urn contains 1 red ball, 4 white balls. Pick an urn randomly. Randomly select a ball from that urn. Without replacing the 1st ball, select a ball from the urn whose color matches the first ball. Q. Make a tree diagram, complete with probabilities describing this situation. Q. Find the probability that the first ball is white. Q. Find the probability that the second ball is white....
2, Five white balls and five black balls are distributed in two urns in such a way that each urn contains five balls. At each step we draw one ball from each urn and exchange them. Let Xn be the number of white balls in the left urn at time n. Compute the transition probability for X,
L. An un contains n red balls and n black balls. Balls are drawn sequentially from the urn one at a time withont replacement. Let the first black ball is chosen. Find EX X denote the number of red balls removed befor
8) For Polya's urn model with r red balls, b black balls and parameters cE N (the number of extra balls we insert every time) and n EN,n 2 3 (the number of times we perform the a) Show that the probability that the k-th chosen ball (k e (1,...,n]) is red is equal to b) Given that the second chosen ball was red compute the probability that the first one experiment), do the following: was red as well.
4. Urn A contains 3 red and 3 black balls, whereas urn B contains 4 red and 6 black balls. If a ball is randomly selected from each urn, what is the probability that the balls will be the same color?
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