You have N boxes (labeled 1,2,..., N), and you have k balls. You drop the balls...
3. You have N boxes (labeled 1,2,... , N), and you have k balls. You drop the balls into the boxes, independently of each other. For each ball the probability that it will land in a particular box is 1/N. Let Xi be the number of balls in box 1 and XN the number of balls in box N. Calculate Corr(X,XN)
3. You have N boxes (labeled 1,2,..., N), and you have k balls. You drop the balls into the boxes, independently of each other. For each ball the probability that it will land in a particular box is 1/N. Let Xi be the number of balls in box 1 and Xv the number of ball in bax N. Calculate Com,X)
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,...
6) Assume that we have n boxes and each one of them contains k white balls and n-k black balls. We choose a box at random and we choose two balls from it (after choosing the first one we are not allowed to put it back). Compute the probability that both balls are white.
6) Assume that we have n boxes and each one of them contains k white balls and n-k black balls. We choose a box at random and we choose two balls from it (after choosing the first one we are not allowed to put it back). Compute the probability that both balls are white.
boxes, n 2, and for each k E {1,...,n} the k-th box contains k watches. In We have every box each watch is defective with probability independently of the other watches in the box. We choose a box randomly. Given that there are no defective watches in it compute the probability that this was the second box boxes, n 2, and for each k E {1,...,n} the k-th box contains k watches. In We have every box each watch is...
5. Three boxes are numbered 1, 2 and 3. For k 1, 2, 3, box k contains k blue marbles and 5 - k red marbles. In a two-step experiment, a box is selected and 2 marbles are drawn from it without replacement. If the probability of selecting box k is proportional to k, then the probability that two marbles drawn have different colours is 6. Two balls are.dropped in such a way that each ball is equally likely to...
5. Three balls are placed at random in three boxes, with no restriction on the number of balls per box. (a) List the 27 equally probable outcomes of this experiment. Be sure to explain your notation. (b) Find the probability of each of the following events: A: "the first box is empty" B: "the first two boxes are empty". C: "no box contains more than one ball". (c) Find the probabilities of events A, B and C when three balls...
Exercise 2.19. We have an urn with balls labeled 1,..., 7. Two balls are drawn. Let X1 be the number of the first ball drawn and X2 the number of the second ball drawn. By counting favorable outcomes, compute the probabilities P(X 1 = 4), P(X2 = 5), and P(X1 = 4,X 2 = 5) in cases (a) and (b) below. (a) The balls are drawn with replacement. (b) The balls are drawn without replacement. (c) Does the answer to...
b) You have three boxes. In the first box, there are 4 white balls and 2 red balls. In the second box, there are only 10 red balls and in the third box, there are 10 red balls and 5 white balls. A red ball is extracted blindly from a box chosen at random, but you don’t know from which box the ball was extracted What is the probability that the first box was chosen? What is the probability that...