 Hello and welcome to the session. In this session we will apply the general multiplication zone in a universal probability model and the rule is probability equivalent A into section B and we can say probability equivalent A and B is equal to probability equivalent A into conditional probability of occurrence of event B given that event A occurs which is equal to probability of event B into conditional probability of occurrence of event A given that event B occurs. Now in our earlier sessions we have discussed conditional probability of occurrence of event A given that event B occurs is equal to probability of event A into section B upon probability of event B where probability of event B is greater than 0. Now let us denote this as equation number 1. Similarly, conditional probability of occurrence of event B given that event A occurs is equal to probability of event A into section B upon probability of event A where probability of event A is greater than 0. Now let us see equation number 2. From equation 1 we have probability of event A into section B is equal to probability of event B into probability that is conditional probability of occurrence of event A given that event B occurs. Now from equation 2 we have probability of event A into section B is equal to probability of event A into conditional probability of occurrence of event B given that event A occurs. So, combining these two results we have probability of event A into section B is equal to probability of event B into conditional probability of occurrence of event A given that event B occurs and this is equal to probability of event A into conditional probability of occurrence of event B given that event A occurs and this is called multiplication rule A and B are dependent in the probability that two events will both happen equals the probability or chance that the first will happen multiplied by the probability that the second will happen knowing that the first has happened. Now let us see how and where we can use multiplication rule. Here let us see this illustration. Now on that contains 10 red balls and 5 blue balls. Two balls are drawn one by one without replacement. What is the probability that first ball drawn is red and second is blue? Now let us start with a solution. Now here we have a rule that a ball contains 5 blue balls and 4 red balls. Now when the ball was drawn is equal to 4 red balls plus 5 blue balls that is 4 plus 5 is equal to 1 but when we do second ball we have a condition that one red ball has already been drawn. We do second ball then remaining number of balls in the back is equal to 3 red balls as we have already drawn a red ball, 3 red balls plus so the remaining number of balls is equal to 8. Now let us define events. Event A is drawing red ball and event B is drawing blue ball. Now we have to find the probability that the first ball drawn is red and second is blue. It means we have to find probability of event A,B. Now from the multiplication rule we know that probability of event A and B is equal to probability of event A into conditional probability of occurrence of event B given that event A occurs that is probability of event A and B is equal to probability of drawing a red ball into conditional probability of drawing a blue ball given that red ball is already drawn. Now here probability of drawing a red ball that is probability of event A is equal to number of outcomes favorable for event A that is 4 upon total number of outcomes that is 4 plus 5 which is equal to 9. So probability of event A is equal to 4 upon 9. We have to find conditional probability of drawing a blue ball given that red ball is already drawn. It means we have to find conditional probability of occurrence of event B given that event A occurs. Now here we have already drawn a red ball it means remaining balls in the bag is equal to 3 red balls plus 5 blue balls that is 3 plus 5 is equal to 8. Now there are 5 blue balls we have to draw a blue ball given that a red ball is already drawn. So this probability that is this conditional probability is equal to 5 upon so probability of event A and B is equal to 4 upon 9 into 5 upon h. Now 4 into 2 is and this is equal to 5 upon 9 into 2 that is 18. So probability of event A and B is equal to 5 upon 18 that means probability that the first ball is red and second is blue is equal to 5 upon 18. Now the other multiplication rule is for dependence when the two events are independent then the multiplication rule is probability of event A and B is equal to probability of event A into probability of event B. Now we can write probability of event A intersection B as probability of event AB where AB means A and B. Now about we have discussed multiplication rule for only two events now we can extend this rule for more than two events. Now consider the same example suppose we have to draw 3 balls successively without representation and we only have to find probability that red ball is red blue and red in order. Now let us define events that event R is red ball, red ball and event B is drawing. Now we have to find probability of drawing a red then blue and then a red ball is equal to probability of drawing a red ball into probability of drawing a blue ball given that red ball is already drawn into probability of drawing a red ball given that red and blue balls are already drawn. So this is equal to probability of event into conditional probability of event B given that event R occurs into conditional probability of occurrence of event R given that event R B occurs. Now here event RB means event R and B and you can say event R intersection B. Now probability of drawing red ball is equal to 4 upon 9 probability of drawing a blue ball given that red ball is already drawn is equal to 5 upon 8. Now the remaining balls when one red and one blue ball is only drawn is equal to 3 over blue that is equal to 7 times probability of drawing a red ball given that red and blue balls are only drawn is equal to 3 upon the remaining balls left in the bag that is 7. So this conditional probability is equal to 3 upon 7. So putting these values here probability of drawing a red ball then a blue ball and then a red ball is equal to 4 upon 9 into 5 upon 8 into 3 upon 7. Now 4 and 3 into 3 is 9. So this is equal to 5 upon 3 into 2 into 7 that is 42. So this is the required probability and in this session we have discussed multiplication rule in probability. So this completes our session. Hope you all have enjoyed this session.