 Hello and welcome to the session. Let us discuss the following question. It says which of the following cannot be valid assignment of probabilities for outcomes of sample space S omega 1, omega 2, omega 3, omega 4, omega 5, omega 6, omega 7. These are the assignment given to us. To which of these are not valid assignment we need to know some basic facts of the probabilities. If S is the sample space having outcomes say omega 1, omega 2, so on, omega n then any assignment is valid. Probability of each omega i is less than equal to 1 and it is greater than equal to 0 that is probability of each omega i lies between 0 and 1 for each omega i belonging to S and the sum of the probabilities should be 1 that is probability of omega 1 plus probability of omega 2 so on probability of omega n is equal to 1. So this knowledge will work as key idea. Let us now move on to the solution and let us see the first assignment omega 1 has probability 0.1, omega 2 has probability 0.01, omega 3 has probability 0.05, omega 4 has probability 0.03, omega 5 has probability 0.01, omega 6 has probability 0.2 and omega 7 has probability 0.6. Now we see that each probability lies between 0 and 1 that is it is greater than equal to 0 and less than equal to 1 and the second condition is that the sum of the probabilities is 1. So P omega 1 plus probability of omega 2 plus probability of omega 3 plus probability of omega 4 plus probability of omega 5 plus probability of omega 6 plus probability of omega 7 and the sum of the probabilities is 1. So this is a valid assignment. Let us now see the second assignment here. The probability of each omega i is 1 by 7 which lies between 0 and 1. The first condition is satisfied and the sum of the probabilities that is probability of omega 1 plus probability of omega 2 plus probability of omega 3 plus probability of omega 4 plus probability of omega 5 plus probability of omega 6 plus probability of omega 7 that is 1 by 7 plus 1 by 7 plus 1 by 7 plus 1 by 7 and the sum is 1. So the second condition is also satisfied so the assignment is valid. Let's now see the third assignment here. The probability of each omega i lies between 0 and 1. So the first condition is satisfied. Now we see the sum of the probabilities is 1 or not and the sum of the probabilities is 2.8 which is not equal to 1. So the second condition is not satisfied. So this cannot be a valid assignment. Now see the next assignment. Now we know that for assignment to be a valid assignment for a sample space as having outcomes omega 1, omega 2, so on omega n, the probability of each omega i should lie between 0 and 1. That is it should be always greater than 0. Now here we see that probability of omega 1 and probability of omega 5 are negative. So the first condition is not satisfied and here we don't need to check the second condition. Because the first condition is not satisfied so we can say that assignment is not valid. Let's now see the next assignment. Now again we know that for an assignment to be a valid assignment probability of each omega should lie between 0 and 1. But here we see that the probability of omega 7 it is equal to 15 upon 14 which is greater than 1. But probability of each omega i should be less than 1. So this cannot be a valid assignment since the first condition is not satisfied. So here we see that the first two assignments are valid and third, fourth and fifth assignments are not valid. Hence yes a is the valid assignment, b is the valid assignment, c is not a valid assignment, d is not a valid assignment and e also not a valid assignment. And this completes the question. Bye for now. Take care. Have a good day.