Box P has 2 red balls and 3 blue balls and box Q has 3 red balls and 1 blue ball. A ball is selected as follows: (i) select a box (ii) choose a ball from the selected box such that each ball in the box is equally likely to be chosen. The probabilities of selecting boxes P and Q are 1/3 and 2/3, respectively. Given that a ball selected in the above process is a red ball, the probability that it came from the box P is:

A point is randomly selected with uniform probability in the X-Y plane within the rectangle with corners at
(0, 0), (1, 0), (1, 2) and (0, 2). If p is the length of the position vector of the point, the expected value of $${p^2}$$ is

Two n bit binary stings, S1 and, are chosen randomly with uniform probability. The probability that the Hamming distance between these strings (the number of bit positions where the two strings different) is equal to d is

An examination paper has 150 multiple-choice questions of one mark each, with each question having four choices. Each incorrect answer fetches-0.25 mark. Suppose 1000 students choose all their answers randomly with uniform probability. The sum total of the expected marks obtained all these students is