 Hi friends I am Purva and today we will discuss the following question. An urn contains 5 red and 2 black balls. 2 balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X? Is X a random variable? Let us begin with the solution now. Now in the question we are given that an urn contains 5 red and 2 black balls and 2 balls are randomly drawn. So let S be the sample space of the experiment of drawing 2 balls. Then S is a set which consists of the following 4 outcomes. The first outcome is that both the balls drawn are red. Second outcome is that the first ball drawn is red and second is black. Third outcome is that the first ball drawn is black and second is red and the fourth outcome is both the balls drawn are black. Now here R represents red ball and B represents black ball. Now we are given that X represents the number of black balls. Then there are 3 cases. First case is when neither of the 2 balls are black that is when both the balls are red. Second case is when one ball is black and one is red and third case is when both the balls are black. Now when neither of the 2 balls are black then we have X is equal to 0 because we are given that X represents the number of black balls. Now when one ball is black and one ball is red then we have X is equal to 1 and when both the balls are black then we have X is equal to 2. So the possible values of X are 0, 1, 2. Now we have to tell that whether X is a random variable or not. Now by the definition of a random variable we know that a random variable is a real valued function whose domain is the sample space of a random experiment. Now here the random experiment is drawing 2 balls and here X represents the number of black balls and there are 4 possible outcomes of drawing 2 balls. So if X represents the number of black balls then for each outcome its value is given as. Now the first outcome is drawing 2 red balls so X of r r is equal to 0 because X represents the number of black balls. Second outcome is first ball is red and second ball is black so X of r b is equal to 1. Third outcome is that the first ball is black and second is red so X of b r is equal to 1 and the fourth outcome is that both the balls are black so X of b v is equal to 2. Now here clearly we can see that X is a real valued function whose domain that is r r, r b, b r and b b is the sample space of the experiment. So by the definition of a random variable we can say that X is a random variable so we write our answer as the possible values of X are 0, 1, 2 and yes X is a random variable. Hope you have understood the solution. Bye and take care.