 Hello and welcome to the session. In this session we will discuss the question which says that a card is chosen at random from a standard deck of 52 cards. Find each probability. First is card drawn is screened given that it is red. Second is card drawn is a hurt given that it is an ace. And third is card drawn is a spade given that it is black. Now before starting the solution of this question we should know a result. And that is if A and B are two events then conditional probability of occurrence of event A given that B appears is equal to probability of event A intersection B upon probability of event B where probability of event B is greater than 0. Now this result will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Now we have given standard deck of 52 cards. In first part we have drawn a king given that the card drawn is red in color. Now we want to find probability P that card drawn is king given that it is red in color. Now let us define the events. Now event A is a king is drawn red in color part is drawn. That means we have to find condition of event A given that event B occurs. Now using the result which is given to us in the key idea the required conditional probability will be equal to probability of event A intersection B upon probability of event B. B is greater than 0. Now we know that this is event A and this is event B. Option B of red color section B contains outcomes that are common to both events A and B. Now we have to find probability P of event A intersection B. Now we know that in a standard deck of 52 cards there are 26 red cards and 26 black cards and these 26 red cards consist of 13 hearts and 13 diamonds and these 26 black cards consist of 13 spades and 13 clubs. Also we know that in a deck there are 4 kings but only are red in color. So number of probability outcomes for event A intersection B is equal to that is 2 red color kings total number of outcomes is equal to 52 that is 52 cards in a standard deck. So event A intersection B is equal to number of probability outcomes for event A intersection B that is 2 upon total number of outcomes that is 52. Now we have to find probability of event B. Now event B is red color card is drawn. Now we know that there are 26 red color cards in a deck. So number of probability outcomes for events B is equal to 26 that is 26 red color cards and total number of outcomes is equal to 52. So probability P of event B is equal to 26 upon 52 which is equal to 1 upon 2. Now using this formula let us find this additional probability. Now this is probability of event A intersection B which is equal to 1 upon 26 probability of event B which is equal to 1 upon 2. So conditional probability of occurrence of event A given that event B have probability of event A intersection B that is 1 upon 26 probability of event B that is 1 upon 2 this is equal to 2 upon 26 this is equal to now we know that 2 into 13 is 26. So this is equal to 1 upon 13. So probability that card drawn is given that it is red in color is equal to 1 upon 13. Now let us start with the second part. In the second part we have to find probability P where card drawn is a heart given that is defined event A card drawn is heart and event B. So we have to find conditional probability of occurrence of event A given that event B occurs. Now here we know this is event A and this is event B. So event A intersection B will be an ace of hearts is drawn. Now we have four cards in a deck of 52 cards and there will be one. So number of favorable outcomes for event A intersection B is equal to 1. So probability P of event A intersection B is equal to 1 upon 52. Now let us find probability of event B now we know that in a deck of 52 cards there are four aces. So number of favorable outcomes for event B is equal to 4. So probability P of event B is equal to 4 upon 52. Now let us find conditional probability of occurrence of event A given that event B occurs. Now this is equal to probability of event A intersection B that is 1 upon 52 upon probability of event B that is 4 upon 52 and on solving. This is equal to 1 upon 4. So probability that card drawn is a heart given that event A is 1 upon 4. Now let us start with the third part here we want to find the probability that card drawn is a spirit given that it is black in color. Now let event A is card drawn is a spirit and event B is card drawn is black in color. So here event A intersection B will be card drawn is a state black color. Now we know that there are 13 states which are always in black color. Number of favorable outcomes for event A intersection B is 13. So probability P of event A intersection B is equal to 13 upon 52 and this is equal to 1 upon 4. Now this is event B that is card drawn is black in color. Now we know that in a deck of 52 cards there are 26 black color cards. So number of favorable outcomes for event B is equal to 26. So probability P of event B is equal to 26 upon 52 and this is equal to 1 upon 2. So conditional probability of occurrence of event A given that event B occurs is equal to probability of event A intersection B that is 1 upon 4 upon probability of event B that is 1 upon 2. So this is equal to 2 upon 4 and this is equal to 1 upon 2. So probability that card drawn is a spirit given that it is black in color is 1 upon 2. So this is the solution of the given question. That's all for this session. Hope you all have enjoyed the session.