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Exercise 1.49. An urn has n -3 green balls and 3 red balls. Draw (balls with...
Exercise 1.49. An urn has n 3 green balls and 3 red balls. Draw&balls with replacement....
Exercise 1.49. An urn has n 3 green balls and 3 red balls. Draw&balls with replacement. Let B denote the event that a red ball is seen at least once. Find P(B) using the following methods. (a) Use inclusion-exclusion with the events Ai = {ith draw is red) Hint. Use the general inclusion-exclusion formula from Fact 1.26 and the binomial theorem from Fact D.2 (b) Decompose the event by considering the events of seeing a red ball exactly k times,...
2. An urn contains two green balls and three red balls. Suppose two balls will be...
2. An urn contains two green balls and three red balls. Suppose two balls will be drawn at random one after another and without replacement (i.e., the first ball is not returned to the urn before the second one is drawn). (a) Find the probabilities of the events A-I A green ball appears in the irst draw (Note, in event B, the first draw is supposed unknown, for example, after the first draw,you do not look at what color the...
An urn contains 5 red balls, 4 green balls, and 2 yellow balls. Draw 3 balls...
An urn contains 5 red balls, 4 green balls, and 2 yellow balls. Draw 3 balls with replacement (draw a ball, record the color, and put ball back before drwing again). What is the probability that your draw (a) consists of all red balls? (b) consists of all the same color? (c) consists of all different colors? (d) consists of at least one green ball? (e) consists of exactly two green balls and one red ball?
1. An urn contains 4 red balls, 3 black balls and 2 green balls. We draw...
1. An urn contains 4 red balls, 3 black balls and 2 green balls. We draw two balls at random (without replacement). If at least one of the two balls is red, we draw one more ball and stop. Otherwise, we draw two more balls without replacement. (i) Compute the probability that the last bal is red (NOTE that for this entire question, your notation is at least as impor tant as your final numerical answer. So, for example, do...
An urn contains four red balls, six black balls, and five green balls. If two balls...
An urn contains four red balls, six black balls, and five green balls. If two balls are selected at random without replacement from the urn, what is the probability that a red ball and a black ball will be selected? (Round your answer to three decimal places.)
Urn A contains 5 green and 3 red balls, and urn B contains 2 green and...
Urn A contains 5 green and 3 red balls, and urn B contains 2 green and 6 red balls. One ball is drawn from urn A and transferred to Urn b. Then one ball is drawn from urn B and transferred to urn A. Let X=the number of green balls in urn A after this process. List the possible values for X and then find the entire probability distribution for X.
An urn contains 5 red balls and 2 green balls. Two balls are drawn one after...
An urn contains 5 red balls and 2 green balls. Two balls are drawn one after the other with replacement. What is the proba- bility that the second ball is red?
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of...
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that...
An urn contains 6 red, 9 green, and 11 blue balls. The following is repeated 3...
An urn contains 6 red, 9 green, and 11 blue balls. The following is repeated 3 times: a ball is selected from the urn at random and removed (called “sampling without replacement”). Give your answers to 3 significant digits. (a) What is the probability that all 3 selected balls are the same color? (b) What is the probability that all 3 selected balls are different colors? (c) Repeat part (a) assuming “sampling with replacement”. That is, the following is repeated...
An urn has 12 red balls and 8 green balls. Pick 5, with replacement. What is...
An urn has 12 red balls and 8 green balls. Pick 5, with replacement. What is the probability of P(3R,2B)? -I understand how to do the problem however I do not get how we can figure out the total # of events is going to be 20^5
Exercise 1.49. An urn has n 3 green balls and 3 red balls. Draw&balls with replacement. Let B denote the event that a red ball is seen at least once. Find P(B) using the following methods. (a) Use inclusion-exclusion with the events Ai = {ith draw is red) Hint. Use the general inclusion-exclusion formula from Fact 1.26 and the binomial theorem from Fact D.2 (b) Decompose the event by considering the events of seeing a red ball exactly k times,...
2. An urn contains two green balls and three red balls. Suppose two balls will be drawn at random one after another and without replacement (i.e., the first ball is not returned to the urn before the second one is drawn). (a) Find the probabilities of the events A-I A green ball appears in the irst draw (Note, in event B, the first draw is supposed unknown, for example, after the first draw,you do not look at what color the...
1. An urn contains 4 red balls, 3 black balls and 2 green balls. We draw two balls at random (without replacement). If at least one of the two balls is red, we draw one more ball and stop. Otherwise, we draw two more balls without replacement. (i) Compute the probability that the last bal is red (NOTE that for this entire question, your notation is at least as impor tant as your final numerical answer. So, for example, do...
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that...
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