 Hi friends, I am Purva and today we will work out the following question. A dime marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event, the number is even and B be the event, the number is red. R, A and B independent. Let E and F be two events. Then E and F are independent. If probability of E intersection F is equal to probability of E into probability of F. So this is the key idea behind our question. Let us begin with the solution now. Now let S be the sample space of the experiment. Now when a die is thrown, we can get six outcomes. That is either 1, 2, 3, 4, 5 or 6. Now it's been given that 1, 2 and 3 are marked in red and 4, 5 and 6 are marked in green. So we get S to be a set which consists of these six outcomes. Now we are given that A is the event that the number is even and B is the event that the number is red. Then we get probability of A is equal to 3 upon 6 that is out of these six outcomes. These are the three outcomes which contain an even number and we get this is equal to 1 upon 2 and probability of B is the probability that the number is red and this is equal to 3 upon 6 because out of these six outcomes these are the three outcomes of the number being red and we get this is equal to 1 upon 2. Now A intersection B is a set which consists of all those outcomes which are both in A and in B. Now A is the event that the number is even and B is the event that the number is red and out of these six outcomes this is the only outcome that has an even number and red. So we get A intersection B is a set which consists of the outcome 2R. Thus we get probability of A intersection B is equal to 1 upon 6 because A intersection B has only one outcome and total number of outcomes are 6. Now probability of A into probability of B is equal to probability of A is equal to 1 upon 2 into probability of B is equal to 1 upon 2 and we get this is equal to 1 upon 4. Now we mark this as 1 and we mark this as 2. So from 1 and 2 we have probability of A intersection B is not equal to probability of A into probability of B. Now by key idea we know that for two events E and F if probability of E intersection F is equal to probability of E into probability of F then E and F are independent. Thus from the key idea we conclude that the events A and B are not independent. Thus we write our answer as A and B are not independent. Hope you have understood the solution. Bye and take care.