ISACA COBIT 5 – Measure (BOK V) Part 11

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• January 27, 2023
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35. Probability Basic concepts – Part 1(BOK V.E.1)

Hey, welcome to this section on basic probability. Before we go into details of that, let’s understand two definitions of probability. Let’s go with classic model first. So the definition of probability is the number of outcomes in which the event occurs divided by the total number of possible outcomes of an experiment. I’m sure since you are doing this course you would definitely be aware of probability concept. But if not, let me make it simple for you. Let’s take an example of a dice. So you have a dice which has six surfaces and on each surface you have number from one to six.

So let’s say this is one here, there are two dots here and then there are three dots here. Four, five, six are on the back of this. So every time you roll this dice, one of these six numbers will come at the top and that’s the number. So when I ask you what is the chance or what is the probability of getting number two in roll of a dice? So if I look at the definition of this, the definition is number of outcomes in which the event occurs. So when I am looking for probability of getting two so there is only one chance when the required outcome will happen divided by the total number of possible outcomes.

So total number of possible outcomes are six because dice could roll anything from one to six. So one divided by six. So this is the probability of getting number two and that’s how we represent probability as p and in the bracket, whatever the result you are expecting or in general terms we can write this the probability of an event is equal to number of expected events divided by total possibilities or the total number of possibilities. So that’s how you could represent the probability.

And what is the probability of getting either one or two? If I say, then you will be saying that probability of one or two will be two. Because there are now there are two chances. Either one can come at the top or two can come at the top, divided by the total number of outcomes, possible outcomes which are six or two by six and all the possibilities which can happen here from one to six, those are represented by sample space. So sample space here is 123456, this is known as sample space.

Another basic thing you need to understand in case of probability is the value of probability at any time will be between zero and one. So the probability can be zero, probability can be one or any number in between that. So zero means that there is no possibility of getting that and one is there are all the possibilities of getting something. So the example of zero probability will be in roll of dice getting number seven. You don’t get number seven when you roll a dice which has number one to six. So the probability of getting seven is zero.

And on the other hand, when I say what’s the probability of getting anything from one to 612345 or six, the probability of that will be one because the outcome will be any of these six numbers only. So six divided by six is equal to one. So probability at any time is going to be between zero and one. So on the previous slide, we talked about the classic definition of probability. The relative frequency of occurrence definition of probability is that probability is equal to number of times an event occurred divided by total number of opportunities for an event to occur.

Let’s say my office starts at 630 and I look at all my past data and I see that in the whole year 40% time I was before 630 and 60% times last year I was either at 630 or after 630. So this is something which I am looking from my past history. So the past history tells that 630 is the time, 06:30 A. m. Is the office time. And before 630, before 630 I was there 40% time and at or after 630 I was 60% of times. So now what is the probability of me reaching office before 630? So based on that, you can say that my probability of getting to the office before 630 is this. So probability which is less than 06:30 A. m. . So reaching there at 06:30 A. m. Would be zero point 40. So this is another definition of probability. So let us look at some of the definitions related to probability. Let’s start with the first definition which is experiment or trial. Many times these are treated as same thing. But there is some minor difference between these. So let’s talk about experiment. So, experiment is something done with an expectation of result. So when we said rolling a dice, rolling a dice was an experiment. So when you roll a dice, you want to see whether there’s a number two coming at the top, that is an experiment.

And trial is the same thing, but how many times? So, if you are rolling dice twice, then you are doing two trials of that experiment. So that’s experiment and trial. The next definition is event or outcome. Event or outcome is the result of that experiment. So the result of that experiment in the rolling of that dice example, we were looking at number two coming at the top. So that’s the event or the outcome of the experiment. And third definition here is sample space. A sample space is set of possible results of that random experiment. So we earlier also talked about that once you are rolling dice, your result could be anything between 126-12-3456 and that’s what is shown as the sample space. So this is the sample space when you are rolling a dice or when you are rolling two dice simultaneously, then your sample space will be something like this. Let’s put it here.

One. One. So both your dices could have one at the top or 1213 and so on up to one, six. And same thing here. Also two, one and up to six, one and six, six. So this will be your sample space when you are rolling two dices at the same time. So these numbers will be 36. There are 36 possibilities. So all these together, if you look at that, six multiplied by six. So these will be 36 possibilities here. That’s your sample space.

36. Probability Basic concepts – Part 2 (BOK V.E.1)

In probability. When you have more than one event, then comes the concept of union and intersection. To understand the concept of union and intersection, we need to understand vein diagram. So let’s understand Wain diagram first and which is Wain diagram. In Wane diagram you will have one rectangle which will represent the total population or all the possibilities. And then you will have circles which will be representing the possibility of a specific event. Let’s say when we took an example of rolling a dice and getting odd number. So all the possibilities in case of dice are 123456. So that is represented by this rectangle. So this rectangle represents 12345 or six. And the event A, our event A was to get the odd number that is represented by a circle. So circle will represent the circle A. This will represent which is A is equal to one, three or five getting the odd number.

So this is event A here. Let’s take another event. Another event was to get any number between one and two. If that was the event B was one or two that will be represented by another circle. So this is event B. And why I show some sort of an overlap here because if you see there’s one number which is common between A and B, which is number one. So that number is represented by this area, the area which is overlapping between A and B, event A and event B. So this is how you represent a vein diagram. Now with this understanding, now let’s look at union and intersection. So let’s start with union. And for that I draw a vein diagram here and going back to the same example of rolling the dice, event A, event A is equal to getting one, three, five, which is the odd number.

And event B was to get a number which is either one or two in a roll of a dice. So how do we represent that? We represent this by A here and event B here, and as we see, there’s a one number which is common between A and B, which is number one. So we put number one in the common area and I put three and five for event A and I put two for event B. So this is how you will represent these two events on the Wane diagram. Now coming to union. So union is represented by this area A. Union B will be one, two, three and five. So this is how you represent union using Wane diagram. Coming to intersection. Same example. Let’s take the same example here. Event A and even B, event A, event B and the common area between these two, that is intersection. And in that common area we just add one number. So we can say that a intersection B is equal to one and a intersection B is equal to one means this area a intersection B satisfies event A and event B and that is the reason when you think of and then think of intersection.

So intersection is and union is or because if you look at these numbers A, union, B, this represent anything, either A or B, everything is included here. But once you look at intersection, this only represents where things are in both A and B and which was number one in our example. So that’s how you represent union and intersection and that’s how you calculate that using a vein diagram. Now let’s look at three types of events in case of probability. Those are mutually exclusive events, independent events and complementary events. And let’s understand these with the help of vein diagram. Let’s talk about mutually exclusive events first.

So going back to the previous example where we draw a vein diagram and we had event A and event B. This is B, this is A and we knew that there is something common. Number one was common between A and B. Mutually exclusive events will be when there is no common thing between these two events. So let’s draw that using vein diagram A and B. So if this is your vein diagram, then these two events are mutually exclusive events. So the example of this could be A being all the odd numbers one, three, five and B being number four. Let’s say B being number four. So there’s nothing common in these. So these two events, event A and event B are mutually exclusive and a intersection B will be nil. There’s nothing which is intersection between A and B because there is no common area between A and B. So that was mutually exclusive events. Coming to independent events.

Two events or more events are called independent events if the occurrence of event A doesn’t change the probability of event B. So when you roll a dice first time, number one comes second time, number five comes when you roll the dice second time. Is that anyway affected by what you got in the first roll? No, because every time you roll a dice those are independent events. Then what is not independent? Let’s take an example of that to understand that. Suppose in a bowl you have three red balls and two blue balls and you pick a ball randomly. So what is the chance that the ball is blue? Okay, you can say that is two divided by total number of balls which are five. So this is probability of getting a blue ball. All right? Now if you don’t replace this ball back and now you draw another ball from the bowl, what is the probability of getting blue ball since you have already drawn a ball earlier? That will affect suppose the first ball which you draw out was a blue ball.

Then the second chance or the second probability will be one divided by four because now there is a one blue ball left in the bowl and three red balls. So that would be the second time when you draw, this will be the probability these two are not the same. So that means if you draw balls from a bowel, then every time you draw without replacement, those are not independent events. Because the probability of the second event depends on what happened in the first event. Coming to the third concept here, which is complementary events. Complementary event is the event which is just opposite of the event A. So the probability that Event A will not occur is denoted by a dash. This is complementary event, so just opposite of Event A. So when you draw a vein diagram, if this was your A, the circle was represented by your event A, then anything which is outside the circle is a dash. So this will be your A dash or the complementary event of Event A. And in this case, as you can guess the total of both A and A dash, the probability of both of these will be one. So probability of A plus probability of a dash is equal to one. So sometimes this complementary event is represented by a dash. At many places you would see this also represented by a bar. But both means the same thing. It’s just the different way of showing complimentary. Event so this completes our discussion on three types of events mutually exclusive, independent and complementary events.

37. Probability Basic concepts – Part 3(BOK V.E.1)

So now coming to two important rules in probability and those are rules of Addition and rule of multiplication. Let’s look at the rule of addition first here. So I have two events, event. Event A is that in roll of dice I get an odd number and the odd numbers are one, three and five. So if I roll a dice I should get one, three and five. That’s one event, event A and event B is getting number one or two. So these are two events, number one and two. Now what rule of Addition says that the probability of event A or Event B is equal to this formula? Let’s understand that. So for that, let’s draw the vein diagram. So here is my overall total population and in that I have event A and Event B. Some part of that is common. Number one is common in both of these and in B I have number two and in A I have number three and five in addition to number one. So this is my vein diagram for these two events. Now when I say probability of A or B, so either A happened or B happened. So that will be probability of a union B. That’s how I will be representing that. So a union B, as we earlier talked, is this whole area. And this whole area represents the probability of any of these events happening, event A or even B. So this is represented by probability of A, plus probability of B, minus probability of A intersection B. Now you would see that things make sense when you want to take the total area of A and B together, you will take the area of a area of B and then you know that a intersection B, which is this area intersection area, this area is common.

When you add these two, this particular area gets added twice. So we need to minus that. And this is what is here minus this. So this is how this fabula has been driven. So if you want to find out the probability of A. So probability of A would be, in this case three by six. Probability of B would be two by six and probability of a union B would be probability of A which is three by six plus probability of B which is two by six. Minus probability of a intersection B and a intersection B is this intersection area which has one possibility which is getting one. In the roll of dice this probability is equal to one by six. So the total probability of either event A or event B happening will be four by six. So here in this case in law of addition, we have one special case. One special case when event A and event B are mutually exclusive. Mutually exclusive means they don’t have anything in common. Something like this, you have a vein diagram where A and B are totally separate.

The example of this could be A is equal to getting, let’s say odd number one, three, five and even B would be, let’s say just getting number two. Number two. So here you can see that there is no overlapping area. Both event A and B are mutually exclusive. So if two events are mutually exclusive, then you can straight away add that. Then you don’t need to subtract a intersection B from that. So here, probability of A union B would be equal to probability of A plus probability of B. Why? Because there is no a intersection B area here. So this is a special case in rule of addition. When two events are mutually exclusive, you can just add the probability of these two events. So this is the rule of addition. On next slide, let us learn the rule of multiplication in probability. Now coming to the rule of multiplication. So in addition, if you remember we were talking about Event A or Event B. Here when we talk of rule of multiplication, we talk of and Event A and Event B happening. So let’s take an example of a bowel. So in this bowl I have six red and four black marbles in that and my Event A is Event A is that first marble is black. So first marble is one marble I pick is that is black. So that’s event A and Event B is the second marble is also black and we are not replacing the first one. So first one when I picked that was black. I’m not putting that back into the bowel.

Then I’m picking the second marble and the second marble is also black. So that’s event b. So second is also black. These are two events and I want to find out the probability of A and B. That means find out the probability that first is black and the second is also black. This is and for that I need to have multiplication. So here is the formula. So probability of a intersection B, that means probability of A and B is equal to probability of A. And what is the probability of A in our example, since there are ten marbles there and we are picking one black and there are four blacks. So the probability of event A happening is four by ten. Because there are four possibilities that the black will be selected and there are total ten marbles in the bowl. Now coming to the second thing, which is probability of B given A has happened. So this is what it mean probability of event B given that A has occurred.

So if A has occurred which is one black has been picked, then in the bowel we are left with nine marbles and out of that three are black. So this probability, which is probability B given A would be three divided by nine and which is equal to on top I will have twelve and 90. So which you can simplify. But this is the probability of a intersection B or A and B. This formula can be simplified when you have two events which are independent. And independent means that event B is not affected by A and that could happen when we do the same experiment but with replacement.

So I draw the first marble, it’s black and I put it back and then again there are ten marbles in the bowl and then I pick the second one. So these two events are independent, nothing is affected by each other. So in that case you can have a simple formula p A, intersection B is equal to probability of A and probability of B. But this is only when A and B are independent. So the same experiment when we do with replacement I pick the first ball or the first marble which is black. The probability of that is four by ten. I replace that, put it back in the bowl and again I picked the second one for that. Also the probability is four x ten. So this will become 16 by 100. So this one is with replacement and this one is without replacement. Without replacement. So this was the rule of multiplication. So earlier we used vein diagram to show the probability and that helped us in finding out the probability. You can also use a tree diagram when you are calculating complex probabilities.

Let’s take two examples. One example of independent event and one example of dependent event. And we earlier said that in independent one event doesn’t affect another event. So let’s say we are tossing a coin and a coin has a head and a tail. So what is the chance that you get a head once you flip a coin? There’s one by two and the probability of getting tail is also one by two. So let’s make a three diagram here. So if I flip a coin head and I could get a tail, the probability of getting head is one by two and probability of getting tail is also one by two. Now I flip the coin second time and second time getting head or tail, what is the probability? Again, these are independent.

And if I ask you what is the probability of getting one head and one tail once you flip a coin, then probably you are looking at these two cases. In both of these cases, you have one head and one tail. So one by four plus one by four is equal to two by four and that means half. So there is a one by two probability that once you flip a coin two times you get one head and one tail irrespective of what comes first. And getting two tails is again one by four. So this was using tree diagram when your events were independent. On next slide, let’s look at an example where events are dependent. So in earlier example of tree diagram, we were looking at an example where events were independent. Now let’s look at the case where events are dependent.

So for that in this example I have ten pieces. So I have ten pieces and I’m selecting out of that. In these ten pieces I have got nine good pieces and I’ve got one defective piece. So that’s my lot. And if I pick one item from that, what is the chance that that item is good? And what is the chance that that item is defective? So let’s draw a three diagram for that. So if I pick the first item, the chance that this is good will be nine by ten. So nine by ten chances are that this item will be good, the first item is good and the first item defective, the chance is one by 10. Now I pick the second item without replacement. Because without replacement means now these are dependent events. So now if I pick the second item and what is the chance that that is good or bad? When the first item was good. So when the first item was good. Now what we have in the box is eight good items and one defective item. So the chance of getting good from this would be eight by nine.

So the chance of getting a good item is eight by nine. And the chance of getting defective in that second pick when the first pick was good, the chance of getting defective is one by nine. And same way if your first item was defective now what you would have in the lot is and now let’s draw a tree here as well. So the first defective, the second good and second defective so the chance that the second item is good is 100% because nine out of nine because there is no defective here. And the chance of getting defective is zero divided by nine, and which is zero. So now let’s look at these two events. So what is the chance that the first item is good and the second item is also good? That we can find out from here. When you multiply nine by ten and eight by nine. So this will be nine by ten multiplied by eight by nine, coming to the second one, the first item good, and the second item defective. The chance of that will be nine by ten multiplied by one by nine, coming to the third, the first defective and the second good. The chance of that is one by ten multiplied by nine by nine.

And the chance that first was defective and second was also defective. What’s the chance of that which is one by 10 multiplied by zero? Because there’s no way you can have two defective first and second, because in the lot itself, there were just one defective. So this becomes zero. So this is how you find out the probability when your events are dependent. And this, as we earlier said, this was shown by formula P. A intersection is equal to probability of A into probability of B given A. So this was the formula here. So either you can use this formula or you can draw this tree diagram to find out the probability when you are doing multiple picks.

38. Probability Distributions – Normal Distribution 1 (BOK V.E.2)

So in probability distribution let’s start our discussion with normal distribution. This is the most commonly used distribution, there are others as well, we will be talking about those other distributions as well but we need to give some more time to normal distribution curve or the normal probability distribution. Let’s look at some of the characteristics of normal distribution or the normal probability distribution. The first thing is that this distribution is symmetrically distributed so let me put the shape of that. So the shape of the normal probability distribution would be something like this. So this is our x axis and this is our y axis. So this is how a normal distribution curve will look like. So the first thing here is it is symmetrically distributed around the mean. So if you look at this portion, the left part and the right part both are equal or the mirror image to each other. The second property of normal distribution is that this has long tail and it is barrel shaped.

So you can look at that that this looks like a barrel shaped and this has a long tail on both sides and when I say long tails, this is infinite tail. This doesn’t meet x axis ever so this meets x axis at infinity so the distance will keep on reducing but this will never touch x axis. And the third property of normal distribution is that mean mode median are same. So this is the point which is the center point, this will be the mean mode or the median. So this is the third property of normal distribution curve. Any normal distribution curve can be defined by two factors one is the mean and the second is the standard deviation. So if you know the mean and the standard deviation you can draw the normal distribution curve. Let’s take few example here, let’s take two normal distribution curves. So let’s say this is curve A and this is curve B and let’s put a vertical axis here at the center. So what do you see here? Curve A and B both have same mean. So mean is same here, and this is that center point. But if you see the curve B is wider so if curve B is wider, let’s say this width compared to the smaller width of A, then we can say that B curve has more standard deviation.

So the standard deviation of curve B is more. On the other hand, if means are different and standard deviation is same, then these curves will be looking like this. So let’s say both have the same standard deviation that means both will have the similar shape or the similar width because width is defined by the standard deviation and the location defines the mean. So this is mean one so this is x bar one and this is x bar two the first one is the mean of the first. Let’s say this is A, this is B-X-1 bar is the mean of curve A and x two bar is the mean of curve B, but both have similar width, so both have the same standard or the similar standard deviation. So on the previous slide we said that any normal distribution curve, standard deviation defines the width, any normal distribution curve, if you look at that, if you look at one standard deviation, both sides from mean.

So let me draw that sample curve here. So this is my normal distribution curve here, this is the mean, let’s say x bar. And if I draw one standard division here, one sigma and minus one sigma, the area which is covered by plus minus one sigma or the one standard division is 68% of the area. So this is what this area will be. So this is going to be 68% of the area. If I go little further, minus two sigma and plus two sigma, this covers around 95% of the area. So let’s use the different color here and two sigma minus and two sigma plus. So this area, if you see the green one, green one will be 95% of the area. And if I go plus minus three sigma, then this is minus three sigma plus three sigma. The area covered by plus minus three sigma is 99. 7% of the area. So this is from here. So this is red one. So red one is 99. 7% and this is for plus minus three sigma. 95 is for plus minus two sigma and 68% is for plus minus one sigma. So this is another property of a normal distribution curve. So let’s look at few more properties of normal probability distribution. The area under the normal curve is equal to one or the 100% and probability of any specific value is zero. And this is true for any continuous distribution. When I say what is the chance that the height of students, which probably I have drawn here, let’s say this is my normal distribution curve for height of students. I see that most of the students have height here, which is around the mean. There are a few students who are very tall.

So this is the tall side of students and this is the short side of students. So this is the normal distribution curve of height of students in a class. The first thing is that area under this curve is one or the 100% and the probability of any specific value is zero. So if I ask you what is the probability that a student height is 170, answer to that will be zero. There is zero possibility or the zero probability of getting exactly the same height. Because this is continuous distribution. When you say 170 CM, then there can be number of fractions below that, 170. 1, 170. 1115. So exactly getting 170 probability of that is zero. That’s another thing you need to remember. Third thing is, as we earlier talked, that plus minus one sigma is, let’s say we talked that this is 68%. So if I go minus one sigma and this side is plus one sigma, the area under these two plus minus one sigma is 68%.

And another thing I said was both the areas left and right are symmetrical. So if it is 68, then one side will be 34. This side 34% area covered on the right side and 34% area covered on the left side. So now if I ask you what is the chance that height of students is more than one sigma. So what I am looking for is more than one sigma is this area. So this area is greater than one sigma. So what is the chance here I can calculate that because now I know if I look at the middle of this curve, right side is 50% and left side is 50%. On right 50% I see that a portion which is from center to one sigma is 34%. So what is left here in this is 50 -34% which is equal to 16% so by knowing few things you can calculate few other things. So here in the third bullet, as I have said that the probability that x is greater than or less than a value is equal to area covered in that direction.

So if I wanted to find out the probability of x greater than one sigma, I can find out the area under that. There are tables for that and there is a formula for calculating this. You will not be using that formula, that much I can assure you. On next slide we will just have a look at that formula how this curve is drawn. But in majority of cases either you will be looking at the table or you will be finding this value using Microsoft Excel or some other statistical tool such as Sigma, Excel, Minitab, etc. But here in this simple case, we could manually calculate that the area under more than one sigma is 16%. So as I talked on the previous slide, here is the formula to draw a normal probability distribution curve which is y is equal to one divided by something. You will not be going into detail of that, but what you need to look here is everything other than sigma which is here and mu which is here.

All other things are constant except y and x. So this equation is in the term of y is equal to something and then the function of sigma and mu. And that’s what we earlier talked, that normal distribution curve is defined by the mean and the standard deviation. So mean and standard deviation will define a normal distribution curve rest all are constants. So what other constants are here? One constant is pi, which is equal to 22 by seven and which comes out to be 3. 14 something and this is a long number. And then there’s another term which is e, another constant which has a value is equal to 2. 71,828. So if you know the sigma and mu, you can draw a normal distribution curve and how this is drawn. So let’s try to draw that. So here I have y axis and here I have x axis. So if you keep on putting a value of x in this equation and the find out equivalent value of y, so you put x is equal to one, then what is the value of y? You find out in this equation, you put x is equal to two and you find out the value of y, et cetera. And once you do that, you can draw the curve and the curve will be looking like this, as we earlier talked. So for a specific value, let’s say at this point at this point, let’s say this is x one and this value is y one and this x one and y one will be related by this specific equation.

Don’t worry about the equation, because this equation you would not be using as a six sigma black palette, you would be using the table or you would be using some software to draw the normal distribution curve or find out the values of probability. So on the previous slide we said that there are tables to find out values when you have a normal distribution curve. But in this world you have number of means and number of standard deviation for a number of things. Let’s say if you are measuring height, so one group will have an average height something and will have a standard deviation something, you look at something else that will have a different mean and different standard deviation. So if these two things define a normal distribution curve, then in this world we will have millions and millions of tables to find out values. But instead of that, what is done is there is a concept of standard normal distribution. So we talked about normal distribution.

Here we are talking of standard normal distribution. So in standard normal distribution, let’s put it here, standard normal distribution, we have tables for this, tables for standard normal distribution. And standard normal distribution has a mean of zero. So when the mean is equal to zero and when the standard deviation is equal to one. So for this we have a table and we can convert anything into normal standard distribution. And how do we do that? We do that by finding the z score or the standard score. And the standard score is equal to x minus mu divided by sigma. Let’s take an example. Suppose we look at a class and we find out that mu is equal to 170 CM, that’s the average height of the population. And we find out that the sigma of that is, let’s say 5 CM, that’s the standard deviation of this population. And now what you want to find out what is the chance that a student height is more than 175 CM. So your x is equal to 175 CM so what’s the chance of that?

So, to draw the normal curve, let’s calculate the z value here. So z value in this case will be 175 -170 which is mu divided by standard division which is five, and this comes out to be one. So z is equal to one. Now, if we draw a normal distribution curve, standard normal distribution curve, here is my standard normal distribution curve. And as I said, standard normal distribution curve has a zero as a mean and one as a standard deviation. So one as a standard deviation. So one, two and three. So this is one sigma, this is two sigma, this is three sigma. Now, when I say z is equal to one. So this is what I am referring to here. Z is equal to one. And when I say I want to find out how many people have height more than 175 centimeter in this, I can look at this area. So this area tells me how many people have height more than 175 centimeters. And as you remember earlier we talked that one sigma is 68% and half of that here is 34% area is this between the mean and one sigma, and the area after one sigma is 16%.

So here what I did was I calculated these values to z score and created a normal standard curve, which is here. And from that I could exactly find out what is the percentage of people who have height more than 175. That came out to be 16% people. So that’s how you use the normal standard table. Here things were simple. Here things were simple because z came out to be one. And we knew for z is equal to one, we knew the value here, that this value is 16% on the right. But suppose if z comes out to be z is equal to 1. 3 or something, then you would need to look at the z table. So once you look at the z table, that will tell you what is the area on the right side of this. So let’s look at one z table and find out. This 16%, which we said here, that when z is equal to one, then the area on the right side is 16%. And then we can even look at some other values when z is something else. So let’s do that on the next slide. So here I have a sample of z table. There could be a number of variation when you look at z table. So the first thing you need to look at is the picture on the top of that z table. So here, what you see here is that when you look at this z table, this is the area this z table is showing somewhere you might have on the left instead of right. So you look at the picture and that will tell you what area this z table is showing. So in this case, exactly. This is what we wanted.

So what we said here is that this area is 16%. When we had one sigma, this is something which we calculated, or roughly calculated previously. So now look at z value of one. So if I go down here, and if I look at 1. 0, and then if I look at zero zero here, then the value of z is coming to be 15. 866. So instead of 16%, this comes out to be 15. 866% and this is the exact value. So earlier when we talked about one sigma is equal to 68%, two sigma is equal to 95% and three sigma is equal to 99. 7%, these were rough values, these are not exact values.

So this is the exact value here. Instead of calculating roughly, if you want to find out the exact value, then instead of 16%, this will be 15. 86%. So if your z comes out to be, let’s say if your z comes out to be 1. 2. So for that you go to 1. 1. So I go here, which is 1. 1 here, and then the next digit is two. So I go to two here and I look at the cross section of this. So I go here and I go here. This is the value which is 00:13 one, three, six. So for z is equal to one point twelve. The probability will be 13. 13 6%. So here, if your z value here was one point, then this area on the right will be 13. 136 percent.

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