 everyone to your third session. Today we're going to do question and answers based on study unit 4. And I've shared with you the weekly session plans. You must please follow that because it's paced in a way that it should help you understand and unpack the the material and be able to follow so that you don't fall behind to help you submit your assignments on time as well. And I've I've I've pasted them in a way that it should help us that we finish with the content a week before this assignment closes so that then it gives you more time if you still want to work on your assignments as well. You can still work on after our last session to make sure that you submit on time. With assignment one remember you will get your third submission right at the end of a week before the closing date. They will give you a a third submission and this is based on mainly because we started eTutorials late but this will not be for all the assignment. It might be but it might not be for all the assignment. It's only for assignment one and that is the reason why I'm saying and because we realize that also many people submitted their assignment one submission one and two one after the other. So we're allowing you to have a third submission because now we relate the message to you to say pace yourself. Don't do submission one and two at the same time. Allow time in between for the both of those two submissions. So now you should be able to do that. Okay so let's continue with today's session which is question and answers for study unit four which is basic probabilities. I've also asked you to go through the content because I post the content before a recording of the content before so that you are able to go through the two hour session or the one hour session of content so that when we come and do the question and answer session we concentrate on how to answer the question to relate into that content that you went through on your own. Right so I'm going to do a quick summary hoping that by half past that should be done with the quick summary and then we will start with the question and answer. If there is something that you don't understand please stop me. Don't raise your hand because I can't see my chat function. Please unmute and let me know that you have a question to ask before I continue and carry on. Don't just write on the chat. If there is a question on the chat somebody just let me do that as well. Okay so when we talk about probabilities we're talking about chances we're talking about the likelihood of something happening so a probability is a chance that an uncertain event will occur and that can always be between the values of zero and one. An impossible event is an event that cannot happen or it does not have any chance of happening and that will always have a probability of zero. An uncertain event is an event that we know for sure that it will happen and it will always have a probability of one. In terms of assessing probabilities as well there are three approaches and usually we use the a priori and empirical probabilities when we work with basic probabilities we don't use the subjective probabilities. You can go and study what all three of them do but a priori is based on the prior knowledge of the process. So if for example creating an event an event is like tossing your coin and when a coin lands on a head I should be able to calculate the probability of a coin landing on a head and that is based on the fact that I know that a coin has a head and a tail and it's a 50-50 chance that it will land on a head and that we calculated by using your x divided by t where x is the outcome that you are getting from that event that you created and your t is your sample space or your grand total or your total number of outcomes all of them combined that will give you your t and also with empirical probabilities it also uses your prior knowledge but that will also you will also have calculated the same way and that will be the number of ways the event can happen which are your outcomes divided by the grand total or the sample space or the number the total number of outcomes that you have and a subjective one is based on a researcher's own opinion so you can have your own opinion in terms of that it might be 70% of a chance that the sun will come out or it's gonna be raining and all that and those who are subjective to the researchers as well. I've already alluded to an event and we know that an event happens when you create an event it's like when you tossing a coin or you pulling a cat or rolling a die or the sun coming up that is an event an event has possible outcomes that can happen so when I roll a die a die has six sides so it can either land on a one or two or three or four five or six those are outcomes so when I do one thing when I do one event it is what we call a simple event rolling a die is a simple event when I do two things rolling a die and tossing a coin I am creating two events and those are called joint events an event that happens and they've got two or more characteristics that will okay and when I roll a die and I toss a coin and if it lands on a head and a die lands on a six I'm creating two joint events for that event that I am creating and a compliment is an event that does not include the others for example with a coin it has two sides a head it's a compliment of a tail a tail is a compliment of a head in case of a die one is a compliment of all the other sides of a die a six is a compliment of all other sides of a die and that's how you will read your compliment event so if I say I roll a die and a die lands on a one what is a compliment of a one it will be all the other events a die landing on a two three four five and a six those are a compliment of a one a sample space is a collection of all events or all outcomes so when you create an event an event is created from this one thing that has outcomes and those outcomes if I combine all of them they create what we call a sample space so a sample space we use that to create an event and within that event it will come out with an outcome like a die it's a sample space because all the sides of a die creates a sample space it creates a grand total the total of all sides and similar to a 52 cut of a cut a deck of a cut it's a sample space if I draw a cut I'm creating an event and the outcome can either be an ace of spade an ace of a diamond and an ace of a hard or a club things like that those are my outcomes okay we can visualize probabilities and events in terms of different things we can use a vain diagram and a vign diagram we can demonstrate simple event like the green cycle is a simple event of a if a represent aces so those will be the cuts that has an ace on it and be representing the cuts that are red and that will be that so we know that in a sample space which is a collection of all the cuts there are cuts that are red there are cuts that are black there are cut there is a cut that is an ace and an ace can be red or black and it's black when it's a club or spade and it's red when it is a heart or a diamond so in between where they both share because let a representing aces and be representing red cut therefore the ace of a diamond and an ace of a heart they are joined event or joint outcomes from ace and red and that creates a joint event as well and that we're going to discuss the joint event and the union events which are your A or B events we will get to that just now we can also visualize events using a contingency table and this is the most easy table to use if you are able to create this table you should be able to represent your event or your probabilities with this table and be able to answer questions and you will see that the majority of questions are based on a contingency table and if there is no contingency table we are going to create one to enable us to answer some of the questions and you can also create your or visualize your events based on a decision tree which also gives you the outcomes as well I'm not going to explain more about this because in that video that I'm explaining I have told you about I said you can go and listen to that or watch that it explains the type of events and what we see in those table in detail simple event and joint probabilities we calculate them by using the outcome satisfying the event divided by the grand total for a simple event so if I want to calculate the simple event like probability of a king I will use outcome satisfying a king divided by the sample space which will be the total of the deck which is 52 and because there are four kings in the deck so there will be four divided by 52 and that will give you your probability a joint event will be if I want to draw a cut that is a king and a spade so I'm creating two events at that point and that will be the number of outcomes satisfying that event which is a king and a spade there is only one cut that is a king and a spade and divide by there grand total or the sample space which will be 52 of them so it will be one divided by 52 when I use the joint event and that is how you will calculate the probabilities always probabilities it's going to be the probability of an event let's use a event a will be the number satisfying divided by the sample space even if it's a joint event event a and b you will realize that yeah I'm using and you can use and and you can use intersect as well they may one and the same thing and that will be observation satisfying that event the joint event divided by the grand total we're going to use that to calculate probabilities there are also what we call mutually exclusive events and those are events that will not happen or cannot happen at the same time and usually this are joint events so the probability of finding the mutually exclusive events event a and b if they are mutually exclusive the probability will be equals to zero because that is an impossible event that will happen we also have what we call collectively exhaustive event which means all events needs to satisfy the sample space so if I have event a which is representing aces and be representing black cards and see representing diamonds and be representing when also hearts event a b c and d will represent in the collectively exhaustive events because all of them make up the sample space but they are not mutually exclusive and ace may be also of a hat so a event a and g will not be mutually extra mutually oh they will not be mutually exclusive because we can have a cat that is an ace and a cat that is an ace with a hat and even b c and d are collectively exhaustive and are also mutually exclusive because a cat cannot be a hat and a black or it can also not be a diamond cat okay this is how you represent joint events and simple events so inside the contingency table those will be your joint event and you can calculate your probability of a joint event there and the grand totals outside or the totals outside that's where you can calculate your simple events and your joint events create what we call marginal probabilities or marginal events that you can use to calculate a simple event because if I add this probability of a one and b one and probability of a one and b two will give me the probability of a and we know that the probability of a it's a simple event a one it's a simple event what we also need to realize is the sum of all probabilities should also be equals to zero so if I add everything that is inside the joint probabilities all these four patterns they should give me one or if I add the total whether I add from the roll b one probability of b one plus probability of b two should give me one or probability of a one and probability of a two should give me one as well in a nutshell what I've just said was that probability is the likelihood of something happening the probability should be between zero and one therefore it means the sum of all probabilities as well that is what we just did now should be equals to one and that also states that if it's equals to one if I need to find the probability of a therefore it means to find in the probability of a we'll have the complement which is one minus the probability of b one minus the probability of b just need to write it correctly one minus the probability of b plus the probability of c and that will give me a complement event we also you need to be aware of the following there is what we call an addition rule but normally they will not tell you that in the exam or in your assignment that apply the addition rule you need to just know that this is part of the addition rule so if we need to find the probability of an event a or event b not both of them but either one of them happening that will be given by the probability of a plus the probability of b minus the probability of a and b and that is how you will find the probability of a or b happening but if and only if the the event a and b are mutually exclusive if they are mutually exclusive then the probability of a and b will be equals to zero because that is an impossible event so the probability of finding a or b will be just given by the probability of a plus the probability of b right only only if they are mutually exclusive otherwise this is the general formula that we're going to be using the probability of a or b will be given by the probability of a plus the probability of b minus the probability of a and b we also have conditional probabilities that is the probability of a happening given that b has already happened the probability that event a is happening given that event b has already happened the keyword here is the given part and that is given by the probability of a and b divided by the probability of the given event which is the b so the conditional probability of probability of a given b it's given by the joint probability of a and b divided by the given probability which is the probability of b and vice versa the probability of b given a it's also the same as the probability of a and b divided by the probability of A. And this probability of A and B, it is not the probability of A and the probability of B. This are not the same. This is a joint probability. It's one value. So if they give you the probability of A and they say the probability of A is 0.5, and they give you the probability of B, which is 0.6, and they ask you to find the probability of A and B. Never ever ever say that probability of A and B is that plus that. They are not the same. These two are three different probabilities. Simple probability and joint probability. Always remember that. So if and only if we are given, or not if and only if, if we are given the probability of a conditional probability, let's say we are given the probability of A given B and we are given the probability of B and they ask us to find the probability of a joint event. If they're asking us, find the probability of A and B. If that is the question, but they have given us the conditional probability, then we're going to apply what we call a multiplication rule, sorry, multiplication rule. Which state that? The probability of A and B is equals to the probability of A given B times the probability of B. Now, remember I told you in the beginning the probability of A is given by X divided by N and I said the probability of A and B is given by observation satisfying the joint event divided by N. Only if you are given the conditional probabilities, then you can find the probability of A and B by using the conditional probability times the probability of B. And that is if they haven't given you the conditional probability, therefore we use the probability of A and B as observation satisfying the joint event divided by N. Right? If and only if A and B are independent, if they tell you that they are independent or you can see that they are independent, then the conditional probability of A given B will be the same as the probability of A and the probability of A, conditional probability of B given A will be the same as the probability of B because the given event has no bearing on what happens to the probability that you are looking for. They are independent, so they do not influence one another only and only if the event A and B are independent. Now, you can also use this to prove if event A and B are independent because if I prove that event A and B are independent, therefore it means the conditional probability of A given B should be the same as the probability of a simple event probability of A. If they are not equal, then they are not independent. If and only if event A and B are independent, then the joint probability of A and B is given by the probability of A times the probability of B and that is multiplication rule. Only if and only if the event A and event B are mutually exclusive, then the probability of a joint event A and B for conditional probabilities is given by the probability of A times the probability of B because we know that the conditional probability of A given B is the same as the probability of A and that is that you need to know about the rules. Now, there are additional other things that you need to know when you work with probabilities and one of those things are the counting rules. So, with counting rules, we want to know the number of outcomes possible outcomes that can happen. Number of ways certain things can be done. So, if one of the k different mutually exclusive and collectively exhaustive event can occur on each of the trials, the number of possible outcomes will equals to k to the power of n. Let's say you are running in a race. In a race, there are seven positions that will get a price. What will be the number of ways that you will get a price and that if in a race seven position and only three of you will get a price and that will be seven to the power of three that will give you that. If you roll a fair die and we know that a fair die has an outcome of six outcomes and if I roll that die three times, how many possible ways I will have the die land on any combination and that will be your six is your outcome trials. There are six of them. So, it will be six k. Sorry, k is your outcomes that are one of them should be mutually exclusive and collectively exhaustive. So, there are six signs and the number of trials are the number of times you roll that die and you're doing it three times that will represent your n. So, it will be six to the power of three and that will give you your outcome that is the counting rule. The other rule is a multiplication rule. It states that if there are k one event of the first trial and k two event of the second trial and up to the other k n event on the nth trial, what will be the number of possible outcomes that can happen and that will be given by the multiplication of all the trials. k one times k two times k three times k four up until k n. If I have to go to a park at a restaurant see a movie and there are three parks for restaurants, six movies, how many different ways or different combinations I can do in order to do all this go to a park, eat at a restaurant and see a movie because there are three restaurants for three parks for a student six movies. So, it will be a multiplication of each trial three times four times six that will give me the different ways I can go and visit all three of them or do all three events and that is multiplication the next one is what we call a factorial. The number of ways an item can be arranged or placed in an order and how many number of ways I can take out the book from a library. If there are four books that I need to take out then I can take out four books or three books two books one book at a time that is your n factorial meaning I can take four times three times two times one that is your n factorial on your calculators and factorials at the other side didn't tell you where to find them because you always use them so this is the power you know how to find the power right on your calculator you do have a function some calculator has an n factorial some calculator has an x factorial and that is the function you're going to use on your calculator to calculate n factorial. So, let's say you want to calculate five factorials so that will be five and then you press the n factorial button on your calculator and then just press the equal sign you will see that you will get a hundred and twenty and that's how you will calculate the number of ways you can take out a book from a library so if here I need to place five books on a bookshelf how many number of ways I can do that I can do it four times five times four times three times two times one and that is 120 ways the last one or not the last one the second last one is what we call a permutation and with permutation it also tells me how many number of ways I can arrange certain things or do certain things if order is a priority or there is an order in how I do things or there is a preference in how I do things you have five books and you are going to put them on the three shelves how many ways they can be ordered on the shelf so they also tell me that I need to put them on the three shelves so those are the orders so you will realize that we do have permutation and then we also have a combination so with permutation always remember that there will be order and preference given how many number of ways can you can the books be ordered on the bookshelf so they're giving me an order in terms of how I must place the books so how can I order them on the day so there is an order on this so it means I need to do a permutation on your calculators I've shown you how to do uh factorials but now I want to say you can use permutation on your calculator go and look for npr or npx on your calculators depending so those who are using a case here uh probably it is on you must look for it it is on a multiplication uh function those who are using a sharp calculator it will be on a number six button so how do we calculate npr on your calculator and in terms of in terms of formula you just use your five as your n and your x is your three the smallest values always your x so how do we arrange these books five factorial divide by five minus three factorial which will be 120 divide by two which is six on your calculator you can press five and press the button that relates to the number which will be second function because it's written in orange or shift depending on your calculator shift or second function and then press your npr button and then press three and press equal and and see if you get the same answer you should get 60 if you're not getting 60 let me know and that is permutation with order combination same as permutation without order if you have to arrange five books are going to be selected to read or you are going to or you have five books and are going to select three to read how many different combinations are you going to uh read those books ignoring the order in which they are being selected so now order is not important to use combination also similar the bigger the number is your n so five is your n and x is your three and you just substitute into the formula five factorial three factorial times five minus three factorial and that will give you ten on your calculator also look for ncr usually it's next to your your permutation it's also written as ncr and on your case it will be on a division on a sharp calculator it will be on button number five and that you do the same you say five and you will press second function or nc or shift or shift and then you will press ncr button that corresponds to that then press three and then press equal and you should get the answer of ten if there is no order there is ten ways you can do things and if there is order there are 60 ways you can do the same thing and those are the counting rules and those are the summary for study unit four that you need to know and learn are there any questions before we start answering questions I know that my summary was long but I needed to do that revision so that we are all on the same page when it comes to doing activities Hi Lizzie, so this is the first few sessions but my top calculator is one that you know all the time because it's up in terms of mode I think six to the power of ten for the previous answer so obviously it's quite logical but once we get to more difficult questions I think I'm on the wrong setting okay you still your your cashier is still set on state mode it gives me a lot of options for the stat mode okay so press mode and then press one for c o m p okay and it should have met at the top okay now then that's simple that's fine yeah all right thank you thank you all right so let's if there are no questions or let me check are there any questions on the chat I'm unable to use my microphone okay all right so there are no questions there so let's do exercises exercise one table one below shows fake news media and the type of personality the postings were about which one of the following statement is incorrect so they give us table 1.1 which has social media and traditional media and celebrity politician and superstar and joint event celebrity and social media there are 1800 simple event celebrity they are 2285 now when when you get a contingency table and you get whole numbers like this they know that these are events events whole numbers probabilities there will be decimals decimals or percentages if you get a table and it has decimals and percentages and those decimals are like zero point are between zero and one zero and one or they are between zero percent and hundred percent then you know that these are probabilities if it's whole numbers then you know that they are events so we're going to apply the formulas for the events so now number one it says and we know that the sample space here is 4000 number one says we need to select which one of the statement is incorrect but because we are evaluating all of them it's fine so number one says find the probability of a joint event c and ss so c is this ss is that find the probability of a joint event c and ss remember the joint probability we can find it event satisfying the event the joint probability divided by the sample space is event c and ss can we find the joint event of those two that is your question are we able to find a joint event like for example a joint event celebrity and sm is 1800 is celebrity and superstar can we find joint events no it's an impossible event because they are from the same variable they say category so any person in this instance it cannot be a celebrity and b a superstar they are mutually exclusive so this is correct because they are mutually exclusive what about the probability of c and sm c and sm the event x and the grand total so calculate the probability of c and sm so that will be given by 1800 divide by 4000 and what is that probability it is no point for five okay so therefore that is also correct the probability of c and tm c and tm it's where they both join so that is 485 divide by 4000 which is not point one two not point one two which is correct the probability so yeah we need to find the probability of sm given that c has already happened so we are asked to find the probability of sm given that c has already happened and we know that is given by the probability of a joint event of both of them so i'm going to do both of them s and c sm and c divide by the probability of a given which is c so what is the joint probability of sm and c sm and c is 1800 remember always divide by the grand total which is 4000 divide by the probability of c z is a that is our x so that will be 2285 divide by 4000 and then here we apply mates right which is 1800 divide by 4000 multiply we change the sign we change the division sign to a multiplication and we flip the second fraction what is at the top goes to the bottom what is at the bottom comes to the top and therefore 4000 and 4000 cancels out and we are left with 1800 divide by 2285 and what is the answer 0.787 so 0.79 that is not 0.79 therefore that is not correct we can also do the same with the second one let's do the second one the second one or the last one says we need to find the probability of sm given s s the probability of social media given that the person is a superstar which will be given by the probability of a joint event probability of joint event sm and s s divide by the probability of s s which is the joint event of s s and sm is 515 divide by 4000 divide by the probability of a simple event s s the input event s is 6 6 6 5 divide by 4000 and we can do the same okay so if I flip and change the whole equation 4000 and 4000 will cancel out and I will be left with 515 515 divide by 6 6 6 5 which is equals to so you apply the same method this i did here which is equals to what is 515 divide by 665 not 0.774 so around down to 0.777 which is that and that's how you will evaluate the statement and find the correct answer any questions yes yes I want to ask something yes you can yeah on number one I'm not sure if my question I will put it the way you would understand it I understand that we we made it as mutually exclusive because of um was there another way besides the reasoning by calculation to prove it as this fall under same category that's what I was wondering uh t and sport let's let's get let's get there c and c and s s can they happen at the same time in terms of the information given can a person be a celebrity and be also a superstar no so that will create what type of an event it's an impossible event right yes and an impossible event has a probability of of zero of zero and we know that a a mutually exclusive event is an impossible event right yes and that is why c and s the probability of them happening at the same time will be equals to zero which is mutually exclusive okay now I understand it from the reasoning point of view it's just I was struggling whether if like you get such a question whereby they're in the same category is it possible to make to prove it under calculation or is it by only reasoning even if you want to prove it by a calculation which value are you going to use because you need for example sm and c have a joint event of 1800 where are you going to get the joint event for c and s s yeah that's that's what I was and that's what I was wondering because it was a bit tricky to prove yes I didn't I choose it as zero that it's mutually exclusive but I was wondering if it happens that I get such a question next time yeah so you just need to look at can I find a value that I can use to calculate and if there is no value therefore it means the event cannot happen so the event if the event doesn't happen therefore the probability of you can't even calculate the probability because it's zero the event is zero all right all right some of the questions might look like this where you are given a contingency table but it's not filled in so what we need to do is before we can even answer the questions we need to complete the entire table so let's complete the entire table and then we will look at what the question is asking us so this is students that we surveyed on a question regarding the means of transport to get to school and the following table gives their summary of the results so we have males and females and bar strain and on a car so how many drives oh how many uses a bus 50 50 50 on a on a car 60 100 100 100 and how many are male 95 95 how many they are female 105 105 and what is the total the grand total 200 it's 200 so now we completed the table happy we can now answer the question we're going to evaluate each and every statement right which one of the following statement is incorrect the probability of f and b which is the probability of female and bus so this is the joint event so we need to find observation satisfying divide by n so how many events are female and bus 20 there is 20 of them divided by 200 which is 200 so 20 divided by 200 is not point one it's not point one so that is correct 25 percent of students uses bus to get to school so yeah they want to know what is the probability of a bus and we can multiply the answer by 100 so that we convert the answer to a percentage so that will be observation satisfying divide by grand total how many uses the bus 15 it'll get less of whether they are female or males so they are 50 divide by 200 and we can multiply that by 100 so that we get it to a percentage and what is the answer 25 percent it's a 25 percent 25 percent so therefore that is correct event m and f are mutually exclusive true that is correct because male and female cannot happen at the same time so that is correct the probability of f o o f o o is 0.735 so we know that the probability of f o o can be given by the probability of f remember we have the probability of a or b or we can write it as the probability of a union b which means one and the same thing it's given by the probability of a plus the probability of b minus the probability of a and b now instead of writing a and v's we're going to use the letters that we are given o and f the probability of f plus the probability of o minus the probability of f and o so what is the probability of f it'll get less of what type of mode of transport that will be 105 divide by 200 remember is the simple event which is x divide by n probability of a always going to be x divide by n the probability of a and b will always be x divide by n for joint event on a contingency table so the probability of o 100 will be 100 over 200 minus the joint event f and o how many events satisfies f and o 59 so that will be 59 over 200 so you can say 105 plus 100 minus 59 divide by 200 what do you get 0.73 that will be 0.73 which that means this is correct if a student is randomly selected the probability that a student is a male given that he uses a train so now we need to calculate the probability of a given right i'm gonna move the space here so we need to find the probability of a male given that they are using a train a train is t given that t therefore we need to calculate the probability of male and train divide by the probability of train probability of male and train male and train which is the joint is 24 divide by 200 divide by the probability of train which is 50 divide by 200 so same if we convert this it will be 24 divide by 200 change the sign to a multiplication flip we're gonna get 200 over 50 and 200 and 200 cancels out that gives us one and this gives us one 24 times 50 divide 24 times one is 24 divide by 50 it's 0.48 which means this is incorrect because the answer is 0.48 and that's how you answer the questions so let's look at more questions which one of the following statement is incorrect with regards to experiments using counting rules and assigning probabilities the number of permutations for four items that can be selected from six items is so they already told you they that you need to use permutation so what do we know about permutation the formula is n pr so always remember the bigger number is your n the smaller number is your r four items which is your r six items which is your n so here you will say six p three four why am i using three which is four or you can use n factorial n minus x factorial so depending on whichever one you want to use and the answer is 360 it's 360 so you can use the this one as well six factorial divide by six minus three factorial i need to also give you 360 the number of combinations also here they give you what you need to be using of five items that can be selected from a group of nine so we have five and nine so we need to use n cr which is n factorial divide by x factorial n minus x factorial or you can use it's nine right nine c five on your calculator by pressing nine and then pressing shift or second function and button number whatever the is it the division or is it on number five and then press five and then press equal what is the answer 126 136 126 and in terms of this you will say nine factorial divide by five factorial times nine minus five factorial and it should also give you one 26 and experiment with four steps and five outcomes possible for each step has 625 outcomes and that is a power right so you will use k n remember your outcomes are k k is your outcome n is your first steps so that will be five to the power of four which is 625 which should be equals to you 625 remember your k is your outcome the outcomes of a sample space okay number four in an experiment with 16 likely outcome each experimental outcome has a probability of in an experiment with 16 likely outcome and each experiment will have the probability of so yeah if i'm gonna use an outcome and i'm gonna assume that my outcome my outcome is a and it should be the number satisfying divide by the sample space each experimental outcome so one of them out of 16 what will be my probability not 0.06 what point not six and not not 0.16 a classical method of assigning probabilities is appropriate when all experimental outcomes are equally likely to happen and this is just an explanation of a empirical probability or an empirical event and that is correct because remember if i have a sample space all of them have to have a likelihood of being selected because like if i toss a coin it might land on a head or it might land on a tail unless i'm tossing an unfair coin which maybe the square coin i don't know but if it's a fair coin all events of all outcomes have an equal chance of coming up when they are being when an event is created as well okay so that will be how you answer the questions relating to probabilities so sometimes the question will look like this you won't have a contingency table the probability of a is equals to 0.4 the probability of b complement the probability of b complement is 0.3 and the probability of a and b is 0.2 which one of the following statement is incorrect so this is different to that because this is the probability of a and this is the probability of a and b uh a complement and b complement this is also different so there are two different this is not the opposite of that right a and b is not a complement of a complement b complement they are different so what i will suggest you do before you answer this question you can draw for yourself a contingency table and on that contingency table so they will be four lines i guess you can have a here and a complement here and have b here and have b complement there and there you will have your totals and they will be your total like i said you can take the information given and create your own contingency table so let's do that we know that the probability of a is 0.4 and we know these are probabilities over it so i know that this is equals to one the probability of a it's in the total is 0.4 the probability of b complement is 0.3 which will be here 0.3 the probability of a and b is 0.2 a and b are there so it's here 0.2 complete the entire table i'm gonna give you two minutes to do that i will be back okay are we done completing the table yes yes but give me the numbers so this will be not comma seven not comma seven and here will be not comma six comma six and this will be not comma two not comma two and yeah not comma one not comma one not comma five not comma five because if i add this should give me zero point four if i add this must give me zero comma seven now i've completed the whole table i can just come here and evaluate because i will just refer to all these values that i see here so number one the probability of a complement and b complement is no point one it's not point one so this is the incorrect one because this should be not point one the probability of b and a complement b and a complement and that is correct the probability of b you get less of a's no point seven zero point seven the probability of a given b that we need to that we need to calculate because then we need to find the probability of a and the probability of b divided by the probability of b the probability of a and b is not point two divide by because these are probability already so we don't have to divide by the grand total so the probability of b is not point seven and the answer zero point two eight five that would be the correct one and the probability of a or b complement is given by the probability of a plus the probability of b complement minus the probability of a and b complement so we did calculate the probability of a no we didn't so the probability of a is zero point four the probability of b complement is zero point three and the probability of a and b complement is zero point two and the answer is zero point four plus zero point three is zero point seven minus zero point two is zero point five and that will be correct and that's how you will evaluate the questions you see a contingency table helps with visualizing the probabilities given or events given and then you can answer the questions easier let's use the eight minutes to look at some of the questions so this is one of those questions and they are telling you here in the beginning assume that event a and b are mutually exclusive and if they tell you that automatically some way you should know that if they tell me that then it means the event a and b are equals to zero so they have given it to you and they also give you that the probability of a is zero point three and the probability of b complement is zero point five so now if you have all that information given in the beginning then it complicates most of these things but not you can also do the same draw up a contingency table because now you have all the other information come on so you have a and a complement b and b complement and here we write totals and total and we know that this should be equals to one probability of a is zero point three the probability of a and b they told us a and b which is here they said they are mutually exclusive so that should be zero therefore it means this probability here is zero point three and here it will be zero point seven and what else we are given the probability of b complement which is here which is zero point five then this will be zero point two and this will be zero point five and therefore this will be zero point five easy can you see that easy to complete this and then once you have a contingency table easy to answer any of these questions which one of the following statement is incorrect so we are told that the probability of a complement then b complement is equals to zero is that true it's false that is incorrect the probability of a and b is equals to zero that's what they told us they are mutually exclusive and we can see from here it is correct and here you just calculate the probability of a plus the probability of b you don't need because this is equals to zero so that it will be how you write your answer the probability of a is zero point three the probability of b is zero point five and that should give you zero point eight and that's how you evaluate the next one the probability of b given a you need to know that that is given by the probability of a and b divide by the probability of b and since the probability of a and b is zero that will be equals to zero as well because they are not independent but they are mutually exclusive and you cannot divide the value by zero so you cannot divide zero by any number it will be equals to zero and that will be correct or actually I wrote it all wrong because this should be the probability of a joint event divided by a because it's b given a but it's still the same because it's the joint event of a and b divided by the given so it's still going to remain the same it will be zero because probability of a joint event is zero then the probability of a and b complement a and b complement is zero point three because that is the block and that's how you will answer the questions easy right you just need to go and practice and practice and on that note we have four minutes you can also it looks like it looks like the same question but they are different yeah they're telling us event a and b are independent so now when they are independent what do we know we know that if we if they are independent there are several things that we need to consider the probability of a given b will be equals to the probability of a and the probability of b given a will be the same as the probability of b that is one of the things that you need to consider when you are answering this so similar you can complete the contingency table I think I am drawing it too big let's draw it too small so we know that we have a and a complement and b and b complement we are given the probability of a which is 0 comma three we are given the probability of b complement which is 0 comma five and we know this is one and we know this is 0 comma seven and we know that this is 0 comma five but we don't know anything in between we cannot do anything unless we do the following what else do we know about independent events we know if we have this scenario we can find because they told us that they are independent so we know that if we need to find the probability of a and b we can use either one of them the probability of a times the probability of b remember conditional probability probability of a given b is given by the probability of a and b divided by the probability of b so if I multiply that will be the probability of a given b times the probability of b is equals to the probability of a and b now I know in terms of independent that is the case so therefore it means this probability of a times the probability of b is equals to the probability of a and b and if I know that then I can find the probability of a and b so let's find the probability of a and b we know what the probability of a is is 0.3 we know what the probability of b is is 0.5 so calculate that 0.3 times 0.5 0.15 it's 0.15 right 0.15 so that is the joint probability then 0.15 we can write it there what is the probability of b and a complement and this 0.5 minus 0.15 0.35 0.35 and you can also do the same this will be 0.35 right because it should give me 70 0.70 0.35 plus 0.35 will give me 0.7 so this will be 0.15 0.15 now I have all the information you can come and answer the question it is your homework I have done enough you can also go and look at this you can also answer same questions so yeah you need to evaluate if they are independent you know when they are independent you it's the same if they are independent it means the conditional probabilities of a given b should be equals to a so you can prove that you can prove that that b given c are independent by using by proving that the probability of b given c should be the same as the probability of b or you can do the probability of c given b you just need to do one of them should be the same as the probability of c if it's independent you also need to prove whether a and b are mutually exclusive they told you that that a and b are that so are they equals to zero if they are equals to zero a and b should be equals to zero under the mutually exclusive event impossible event it means a and c the probability of a and c should also be equals to zero can you prove that if you are able to prove it then they will be impossible event event a and b are dependent dependent it means they are not equal so you can use the same the probability of a given b should not be equals to the probability of b or the probability of b given a should not be equals to the probability of b i wrote the first one wrong it should say the probability of a so if you can prove that that is the case then they are dependent a and b are independent you do the same but they need to be equal you you do the same statement but they will have to be equal because if they are independent then then now you need to choose which one is the correct one by choosing any of these symbols that's how complex it is with multiple choice questions you work three times as much as if you were writing a question where you write answers only also this is how they can also ask you questions you don't also not only have to know the calculations you need to know the theory behind every calculation that is why it's very important that you understand the content as well so i've given you lots and lots of work to do if there are things that you are not sure about we can have a discussion on the whatsapp group remember you cannot post the question without showing us how you worked it out here is another example i'm not going to even give you some hint now i've given you enough here is another example where you need to calculate the probability of a and b here is another example you need to complete the table and then answer the question they are asking pay attention to the words given it means you will need to calculate the probability of something given and the given will be that so given will be the neural NP and here they're asking if it's a boy so it's b given p and then you just need to use the formula to calculate remember these are events not probabilities so you will have to divide by the grand total this is another way that they can ask you questions in a statement format if you are stuck let me know how i can unstuck you this is another way i've given you a lot of questions you can see that you can practice and practice if there is things that you don't understand you can ask on the whatsapp group and that concludes today's session our one hour 30 minutes any questions any comments before we close off the session any final word and without any comment or questions or final word then i will just end right here and say thank you for coming through just to recap please remember that you need to understand and learn what the basic probabilities are all about because we're not going to have another session um on basic probabilities until the last session of a week before you go submit your second submission uh so go through the work understand the events understand how to answer them and also remember that you need to also understand the theory behind the basic probabilities not only the formulas and the questions and as you can see that there are so many formulas that we are using and i just summarized all the formulas when i was doing the summary you can also have them somewhere neatly written so that you can always refer to them because you need to know how to calculate the probability of a and b or the joint probabilities or a simple probabilities you need to know how to calculate the probability of a or b which is either and if a and b are mutually exclusive you need to know that the probability will be zero you also need to know how to find the conditional probabilities and in case where you need to use multiplication rule is when you are given the conditional probabilities and you are asked to find the probability of a and b and if even a and b are independent therefore it means the conditional probability of a given b will be the same as the probability of a and you can use that to prove if events are independent or not and that is what we've learned today have a lovely sunday thank you have a great sunday thank you thank you bye