 Good morning friends. I'm poor one today. We will work out the following question. Let a and b be independent events with Probability of a is equal to zero point three and probability of b is equal to zero point four Find first probability of a intersection b second probability of a union b third probability of a given b and fourth probability of b given a Let e and f be two events then the conditional probability of the event e given that f has occurred that is Probability of e given f is given by probability of e given f is equal to probability of e intersection f Upon probability of f and here we have probability of f is not equal to zero So this is the first formula which we will use in this question The second formula which we will use in this question is let e and f be two events now if e and f are independent then we have 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 we are given that probability of a is equal to zero point three and Probability of b is equal to zero point four We are also given that a and b are independent events Now in the first part we have to find probability of a intersection b Now by key idea we know that if two events e and f are independent then probability of e intersection f is equal to probability of e into probability of f now here We know that a and b are independent so we get probability of a intersection b is equal to probability of a into probability of b This is since a and b are independent Now this implies probability of A intersection B is equal to probability of A which is equal to 0.3 into probability of B which is equal to 0.4 and this is equal to 0.12 so we have got probability of A intersection B as 0.12. Now in the second part we have to find probability of A union B. Now for two events A and B 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 now probability of A is equal to 0.3 so we have 0.3 plus probability of B is equal to 0.4 minus probability of A intersection B is equal to 0.12 by first part we know so we have minus 0.12. So this implies probability of A union B is equal to 0.58. Now in the third part we have to find probability of A given B by key idea we know that for two events E and F the conditional probability of the event E given that F has occurred that is probability of E given F is given by probability of E intersection F upon probability of F. So here we have probability of A given B is equal to probability of A intersection B upon probability of B and here probability of B should not be equal to 0. Now this is equal to probability of A intersection B which is equal to 0.12 we know by first part upon probability of B which is equal to 0.4 this is given to us and this implies probability of A given B is equal to 0.3. Now in the fourth part we have to find probability of B given A. Again by key idea we know that probability of E given F is equal to probability of E intersection F upon probability of F and here probability of F should not be equal to 0. So probability of B given A is equal to probability of B intersection A upon probability of A and here probability of A is not equal to 0. This is equal to probability of A intersection B upon probability of A. Now since A intersection B is equal to B intersection A so we know that probability of A intersection B is equal to probability of B intersection A. Now this is equal to 0.12 because probability of A intersection B is equal to 0.12 we know this from first part upon probability of A is equal to 0.3 this is given to us. So this implies probability of B given A is equal to 0.4. Thus we write our answer as answer for the first part is 0.12 answer for the second part is 0.58 answer for the third part is 0.3 and answer for the fourth part is 0.4. Hope you have understood the solution. Bye and take care.