 Hello and welcome to the session. In this session we will use the 2-way table as a sample space to decide if events are independent and to approximate conditional probabilities. Now, in one of our earlier sessions of presenting data, we were discussed about marginal and conditional distributions using 2-way frequency table. In this session, we will continue with the 2-way tables and we will see how can we decide whether the events are independent or not. We will also learn to find conditional probabilities using 2-way tables. Now, the given 2-way table shows how the students in a class performed. In their first driving test, some took the class to prepare and others did not. Now, we want to find probability that a student selected randomly passed given that he took the class. Now, let us define the events. Event A is took the class and event B is student pass on this table. In this table, we are given class as no class in columns and failed in rows. So, in this table, this is event A versus event B. Now, here in this table, we can see that no class will be event not A. And we can write it as event A bar, this means event A not happening. Also, A bar denotes A complement and here B bar denotes B complement or event B not happening. Now, here we can see number of students who took class or passed is and this gives us frequency of event A intersection B. Also, you can see number of students who took no class is 38 and this gives us frequency of event A complement A. intersection B. Now, number of students who took class 18, this gives us frequency A complement. Now, number of students who took no class, so this gives us frequency of A complement intersection B complement. Now, let us find the total of rows and columns. Now, from table, we have total of row 1 which is equal to 112 when total of row 2 is 18 plus 32 that is equal to 50. Total of column 1 is 64 plus 18 which is equal to 82 and total of column 2 is 48 plus 32 which is equal to 80. Now, adding either total of rows or total of columns. Now, let us add of columns that is 82 plus 8 which is equal to 162. Now, we can also see that total of rows is also 162. Now, to find probabilities from 2-way tables, we first define the events. Now, total of column 1 divided by total students according to event A will give us probability of event A is equal to 82 upon 162. Now, probability of event A complement will be equal to total column 2 divided by total number of students. So, this will be 80 upon 162, total of row 1 will give us probability of event B which is equal to 112 upon 162. Total of row 2 will give us probability of event B complement which is equal to 50 upon 162. Now, we have to find conditional probability that a student is selected randomly given that he took the class. It means we have to find conditional probability that B given that event A occurs. Now, we know that this conditional probability is equal to probability of event A intersection B upon probability of event where probability of event A intersection B means we have to find probability that a student selected randomly took the class. Now, here we know that number of students who took class, so number of students for event A intersection B is equal to 64. Now, here total number of outcomes is equal to 162. So, probability of event A intersection B is equal to number of favorable outcomes for event A intersection B that is 64 upon total number of outcomes that is 162. Now, we already know that probability of event A is equal to 82 upon 162. Now, putting these values in this formula, we have conditional probability of occurrence of event B given that event A occurs is equal to probability of event A intersection B that is 64 upon 162 upon probability of event A that is 82 upon 162. Now, further this is equal to 64 upon 82. Now, we know that 2 into 32 is 64 and 2 into 31 is 82. So, this is equal to 32 upon 31. So, conditional probability that a student selected randomly passed given that he took the class is 32 upon 41. Now, here since total number of outcomes cancelled out, so we may directly quote the frequency values in the formula that is conditional probability of occurrence of event B given that event A occurs is equal to probability of event A intersection B upon probability of event A which is equal to number of students who took class and upon number of students who took class values frequency of event A intersection B upon frequency of event A. Now, from the table you can see frequency of event A intersection B is 64 upon frequency of event A 82. So, here also in solving we see that this is equal to 32 upon 41. Here we can find that a student selected randomly took class given that he failed. This means we have to find conditional probability of occurrence of event given that event A complement occurs. So, this will be equal to probability of event A intersection B complement upon probability of event B complement. Now, frequency of event A intersection B complement is 18 and frequency of event B complement is 50. So, this is equal to 18 upon 50. Now, we know that 2 into 9 is 18 and 2 into 25 is 50. So, this is equal to 9 upon 25. Now, let us check whether events A and B are independent or not. Now, A and B will be independent events if conditional probability of occurrence of event A given that event B occurs is equal to probability of event A or conditional probability of occurrence of event B given that event A occurs is equal to probability of event B. Now, we have this conditional probability of occurrence of event B given that event A occurs is equal to 32 upon 31 and probability of event B is equal to number of favorable outcomes for event B that is 112 upon total number of outcomes that is 162. Now, we know that 2 into 56 is 112 and 2 into 81 is 162. So, this is equal to 56 upon 81. So, here you can see conditional probability of occurrence of event B given that event A occurs is not equal to probability of event B. So, the events A and B are not independent. So, in this session we have discussed how to use the two-way table as a simple space to decide if events are independent and to approximate conditional probabilities. And this completes our session. Hope you all have enjoyed the session.