 Hi friends, I am Purva and today we will discuss the following question. One card is drawn at random from a well-shuffled deck of 52 cards. In which of the following cases are the events E and F independent? First, E. The card drawn is a spade and the event F is the card drawn is an ace. Second part is E is the event that the card drawn is black and F is the event the card drawn is a king. Third part is E is the event the card drawn is a king or queen and F is the event the card drawn is a queen or jack. Now if E and F are two independent events 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 since there are 52 cards so the sample space S will contain 52 outcomes for 52 cards. Now in the first part we are given that E is the event that the card drawn is a spade and F is the event that the card drawn is an ace. So we have probability of E is equal to 13 upon 52 because there are 13 spades in a deck of 52 cards and we have this is equal to 1 upon 4. Now probability of F is equal to 4 upon 52 as there are 4 aces in a deck of 52 cards and we get this is equal to 1 upon 13. Now we have to find whether the events E and F are independent or not. Now by key idea we know that if E and F are two independent events then we have probability of E intersection F is equal to probability of E into probability of F. So here we will find probability of E intersection F that is probability that card is a spade and an ace. Now in a deck of 52 cards there is only one card which is a spade and an ace. So we get this is equal to 1 upon 52. Now probability of E into probability of F is equal to probability of E is equal to 1 upon 4 into probability of F is equal to 1 upon 13 and we get this is equal to 1 upon 52. Now we mark this as 1 and we mark this as 2. So from 1 and 2 we have probability of E intersection F is equal to probability of E into probability of F. So by key idea we get thus the events E and F are independent. Now in the second part we are given that E is the event that the card drawn is black and F is the event that the card drawn is a king. So we get probability of E is equal to 26 upon 52 as there are 26 black cards in a deck of 52 cards and this is equal to 1 upon 2 and we get probability of F is equal to 4 upon 52. That is the probability that the card drawn is a king is 4 upon 52 as there are 4 kings in a deck of 52 cards and we get this is equal to 1 upon 13. Now again to find whether the events E and F are independent or not first we will find probability of E intersection F that is probability that card is black and a king. This is equal to 2 upon 52 because in a deck of 52 cards only 2 black cards are kings and we get this is equal to 1 upon 26. Now probability of E into probability of F is equal to probability of E is equal to 1 upon 2 into probability of F is equal to 1 upon 13 and we get this is equal to 1 upon 26. Now we mark this as 1 and we mark this as 2. So from 1 and 2 we have probability of E intersection F is equal to probability of E into probability of F Now by key idea we know that if E and F are two independent events then we have probability of E intersection F is equal to probability of E into probability of F so we get thus E and F are independent events. Now in the third part we are given that E is the event that the card drawn is a king or queen and F is the event that the card drawn is a queen or jack. So probability of E that is the probability that the card drawn is a king or queen is equal to 8 upon 52 as there are 4 kings and 4 queens in a deck of 52 cards and we get this is equal to 4 upon 26. Now probability of F that is probability that the card drawn is a queen or jack is equal to 8 upon 52 as there are 4 queens and 4 jacks in a deck of 52 cards and we get this is equal to 4 upon 26. Now to find whether E and F are independent or not we find probability of E intersection F that is probability that card drawn is a queen and we get this is equal to 4 upon 52 because there are 4 queens in a deck of 52 cards and we get this is equal to 1 upon 13 now E is the event that the card drawn is a king or queen and F is the event that the card drawn is a queen or jack so probability of E intersection F will be probability that card drawn is a queen now probability of E into probability of F is equal to 4 upon 26 into 4 upon 26 and we get this is equal to 4 upon 169 now we mark this as 1 and we mark this as 2 so from 1 and 2 we have probability of E intersection F is not equal to probability of E into probability of F so by key idea we get the events E and F are not independent thus we get the answer for this question as 1st and 2nd hope you have understood the solution bye and take care