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:

Let $$f(x)$$ be the continuous probability density function of a random variable X. The probability that $$a\, < \,X\, \le \,b$$, is:

A bag contains 10 blue marbles, 20 green marbles and 30 red marbles. A marble is drawn from the bag, its colour recorded and it is put back in the bag. This process is repeated 3 times. The probability that no two of the marbles drawn have the same colour is

What is the first order predicate calculus statement equivalent to the following?
Every teacher is liked by some student