 Welcome to the session. I am Mr. Praveen Yalapa Kumbhar, working as assistant professor in Vulture Institute of Technology. Now in this session we want to study conditional probability. The learning outcome of this session is at the end of this session student will be able to explain the concept of conditional probability. Now the contents of these sessions are conditional probability, independent event and multiplication rule of probability. Before going to this conditional probability we can, probability means what we can understood. Now probability means what probability is equal to the n upon s, n means what number of events and s means number of sample space. Now the first is conditional probability. The probability of an event based on the fact that some other event has occurred will occur or is occurring. The probability of event b occurring given that event a has occurred is usually stated as the conditional probability of b given a that is p p by a. In shortly the conditional probability means what the probability of occurrence of b is purely depends on the occurrence of a means in generalized language we can say that the probability of occurrence of any event is depend on the probability of occurrence of another event that is called as conditional probability. Therefore in mathematical formula the conditional probability p b by a means conditional probability of b in condition given as occurrence of a then is equal to probability of a intersection b divided by probability of a. In mathematical term again that evaluated simplify that is p a and b divided by probability of a. The same way we can calculate that probability of occurrence of a for conditional given as a b then means what the probability of occurrence of a for the condition b that is equal to probability of a intersection b divided by probability of b is equal to probability of a and b divided by probability of b. This is the basic formula for the conditional probability we can use this formula for solving the problems. Now we consider first example a number from the sample space s equal to 2, 3, 4, 5, 6, 7, 8, 9 is randomly selected given the defined events a and b. Now event a is selected number is or and b is selected number is a multiple of 3. Now the question is that find the following probabilities a probability of b, b probability of a and b, c probability b by a means what here first of all we find out probability of b. Now the basic definition now we apply here the basic definition of probability for calculation of a probability b we require first we calculate what is the definition of probability the probability definition is n by s, n means what what is given for the probability for b selected number is a multiple of 3 and divided by s means what number of sample space. Now here if you observe number of sample space is 1, 2, 3, 4, 5, 6, 7, 8 therefore number of sample space is what 8 and the probability whatever we give the n for the number b selected number is a multiple of 3 then you find out in from the sample space that is number of multiple of 3 that is 3, 6, 9 means therefore for the probability b means what selected number is a multiple of 3 means what 3, 6, 9 therefore the n is what 3 and s is what 8 therefore probability of b definition of probability b is equal to n by s. Now n is what 1, 2, 3, 3 divided by s that is 3 by 8. Now similarly for calculation of probability of a and b now we calculate probability a and b now we know that and means what here intersection intersection we use here mathematically now calculate probability a and b now what is the probability first we calculate probability of a and means what intersection then probability b means what probability b then from that intersection intersection means what common element common element of these two events that is a event and b event we calculate. Now suppose now first of all we concentrate on probability of a what is the probability of a probability of a is selected number is r selected number is r from which from the sample space selected number odd is what 3, 5, 7, 9 that is we written here 3, 5, 7, 9 now and means what intersection we give the intersection any problem now the second is b what selected number is a multiple of 3 selected number is a multiple of 3 we calculated here yes 3, 6, 9 we write 3, 6, 9 now here we calculate the intersection intersection between what probability of a and b now calculate the intersection means what common elements now common elements if you observe 3 is a common element in the both yes 3 is a common element 6 is there no 6 is not there 9, 7 is there no 9 is there yes then write down 3 and 9 now total the what is n n become 2 and what is a yes 8 therefore it become as a 2 by 8 is equal to 1 by 4 this is what probability of b a and b. Now the most important part whichever the conditional probability we want to calculate conditional probability in occurrence of a now probability b by a we know the formula we studied in the previous slide that is what probability b a and b divided by probability of a now we calculated probability a and b yes what is that one 1 by 4 you write here 1 by 4 what is the probability of a simply you calculate as term as b we calculate 3 by 8 same you calculate for probability a now that is answer is 4 by 8 1 by 4 divided by 4 by 8 you get 1 by 2 that is what conditional probability answer now consider the second example second example given a family with the two children find the probability that both are the boys given that at least one is a boy now here the conditional probability we use that is probability of b by a is equal to probability of a and b divided by probability of a now what is a yes yes is a sample space sample space both are girl girl boy girl and both are boy a is at least one boy b is both are boys a girl and boy boy girl boy boy and b is boy boy now probability of a and b is equal to probability you calculate and the same concept we use 1 by 4 a 3 by 4 and put the values and we get 1 by 3 now independent events two events are independent if the occurrence of one of them has no effect on the probability of either probability of b by a is equal to probability of b or probability of a by b is equal to probability of a now example a single card is a randomly selected from a standard 52 card as given the defined events a and b is the selected card is s b is the selected card is red find the following probabilities probability of b probability of a and b probability of b by a we calculate the same procedure we use is the previous one and the answer is this one now here multiplication rules for probability if a and b are the two events then probability of a and b is equal to probability of a divided into probability of b by a now recall if a and b are independent events then above formula become as probability of a and b is equal to probability of a into probability of b again the same concept for this example we use a jar contains four red marbles three blue marbles and two yellow marbles what is the probability that a red marble is selected and then a blue one without replacement the same we use for this formulas and without replacement we solve this formula answer become 0.1667 now here the same formula but here one with replacement we use and we calculate probability of red and blue and this is a answer here use the same formulas references thank you