 Good morning friends. I am Purva and today we will work out the following question. Given that the events A and B are such that probability of A is equal to 1 upon 2, probability of A union B is equal to 3 upon 5 and probability of B is equal to P. Find P if they are one mutually exclusive second 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 first formula which we will use in this question. Now the second formula which we will use in this question is if E and F are two events then probability of E union F is equal to probability of E plus probability of F minus probability of E intersection F. So this is the key idea behind our question. Let us begin with the solution now. Now we are given that probability of A is equal to 1 upon 2, probability of A union B is equal to 3 upon 5 and probability of B is equal to P. And in the first part we have to find the value of P if A and B are mutually exclusive. So we are given that A and B are mutually exclusive. Now A and B are mutually exclusive means no element of A and B is common that is A intersection B is equal to 5. So we get probability of A intersection B is equal to 0. Now by key idea we know that for two events E and F probability of E union F is given by probability of E plus probability of F minus probability of E intersection F. So now we know that probability of A union B is equal to probability of A plus probability of B minus probability of A intersection B. Probability of A union B is equal to 3 upon 5 and this is equal to probability of A which is equal to 1 upon 2 plus probability of B which is equal to P minus probability of A intersection B which is equal to 0. Now this implies 3 upon 5 minus 1 upon 2 is equal to P or we can write this as P is equal to 3 upon 5 minus 1 upon 2 which implies P is equal to 1 upon 10. So we have got the value of P as 1 upon 10. Now in the second part we have to find the value of P if A and B are independent. Now by key idea we know that if two events E and F are independent then we have probability of E intersection F is equal to probability of E into probability of F. Now here we have A and B are independent so we get probability of A intersection B is equal to probability of A into probability of B. We mark this as 1. Also we know that probability of A union B is equal to probability of A plus probability of B minus probability of A intersection B. That is probability of A union B is equal to probability of A plus probability of B minus probability of A into probability of B using 1. Now probability of A union B is equal to 3 upon 5. This is equal to probability of A which is equal to 1 upon 2 plus probability of B which is equal to P minus probability of A that is 1 upon 2 into probability of B that is P. This implies 3 upon 5 minus 1 upon 2 is equal to 1 upon 2 into P. Now this implies 1 upon 10 is equal to 1 upon 2 into P and this implies 1 upon 5 is equal to P or we have P is equal to 1 upon 5. So in the second part we have got the value of P as 1 upon 5. Thus we write our answer as for the first part we have P is equal to 1 upon 10 and for the second part we have P is equal to 1 upon 5. Hope you have understood the solution. Bye and take care.