 Hello and welcome to the session I am Deepika here. Let's discuss a question which says two dice are thrown simultaneously if x denotes the number of sixes find the expectation of x. Let us first understand how to find the mean of our random variable. Let x be our random variable whose possible values x1, x2, x3, so on, xn occur with probabilities p1, p2, p3, so on, pn respectively. The mean of x denoted by mu is the number sigma xi pi i varying from 1 to n. Now the mean of our random variable x is also called the expectation of x and this is denoted by ex. So we have ex is equal to mu which is equal to sigma xi pi i varying from 1 to n. In other words the mean or expectation of our random variable x is the sum of the products of all possible values of x by their respective probabilities. So this is the key idea behind our question. We will take the help of this key idea to solve the above question. Now let's start the solution. Now according to the question two guys are thrown simultaneously. So when two guys are thrown the sample space of this experiment has 36 elements. Now the elements of this sample space are ordered pairs where the first digit shows the number on the first die and the second digit shows the number on the second die when two dies are thrown simultaneously. That is the sample space of the given experiment 36 elements. Again we are given if x denotes the number of sixes we have to find the expectation of x. So the random variable x that is the number of sixes on the two dies takes the values 0, 1 or 2 because in these ordered pairs either we have no sixes or we have one six or we have two sixes so the random variable x can take values 0, 1 or 2. So the probability distribution x is when x is 0 or in the sample space there are 25 ordered pairs which do not have any six so when x is 0 px is equal to 25 over 36. Now when x is 1 in the sample space there are 10 ordered pairs which have only one six so when x is 1 px is 10 over 36. Now when x is 2 there is only one pair which have two sixes so when x is 2 probability of x is 1 over 36 so according to our key idea the mean of a random variable x which is also called the expectation of x is equal to sigma xi pi i varying from 1 to 10 so this is equal to 0 into 25 over 36 plus 1 into 10 over 36 plus 2 into 1 over 36 and this is again equal to 0 plus 5 over 18 plus 1 over 18 and this is again equal to 6 over 18 which is equal to 1 over 3 so when two dies are thrown simultaneously if x denotes the number of sixes then the expectation of x is 1 over 3 so this is the answer for the above question this completes our session I hope the solution is clear to you bye and have a nice day.