ASQ Six Sigma Green Belt – Objectives – Hyperledger Part 3

7. Whiteboard – Hyperledger Nodes

What does Rule of Multiplication say? Is that the probability that event A and B both occur? If you want to find out the probability of A and B, then that is equal to probability of A multiplied by probability of event B given A has occurred. So let’s use the Venn diagram to understand this. So here is my Venn diagram. This is the total sample space. Now, here I have event A. And here I have event B. And when I say I want to find out the probability of A and B both happening and what does this mean is this area? This is the probability of both A and B both happening.

Now this is shown by a intersection B and we have talked about union and intersection earlier as well. So probability of a intersection B is equal to probability of A multiplied by probability of B. And this sign means given A has happened. To understand this in a little bit more clarity, we will take two different examples. One example where events are independent and the second example where events are dependent.

And we have talked about independent events earlier as well. Independent means the happening of one event doesn’t affect the other event. And we talked about the example of independent event as rolling the dice. So if we roll a dice one time, the result of that does not affect the roll of dice second time. And opposite of that was dependent even there. We talked about a case where we had two red balls and two blue balls in a bowl, and we were trying to find out the probability of getting two red balls in two successive draws. Let’s look at these examples and that will clarify the Rule of Multiplication. Let’s start with independent events. Here I’m rolling a dice two times and I’m looking at the probability of getting six in both. So here my event A is getting number six. So let me put it here. So A is getting number six and B is also getting number six in the roll of dice.

And as we already discussed that these two events are independent happening of one doesn’t affect the other. And now how do I find out this probability? This probability? I can find out by this formula p A intersection B, which means probability of A and B. A is getting six in the first roll. B is getting six in the second time. I roll the dice is equal to probability of A. As you can see in the formula above, multiplied by probability of B, provided a has happened. And since these are independent event, then it doesn’t matter whether A has happened or not happened. So this will be probability of B and this will come out to be one by six, which is the probability of event A multiplied by probability of B, which is also one by six, is equal to one by 36.

So the probability of getting two six in two rows of dies is one divided by 36. So this was an example of independent event. Now coming to the example of dependent event, let’s see that on the next slide. So here I have that example. Here I have ten candies in the plate. So if you look in the plate on the right side I have ten candies, might be a little bit more, but let’s think that these are ten candies. In this five are green, two are yellow, two orange and one red.

Now I want to randomly pick two of these candies and I want to find out the probability that both of them are yellow. So here what I’m looking at is probability of getting two yellows in two successive picks. So how do I find this out? So this will be probability of a intersection B which is probability of A and B. And here A is getting a yellow candy in the first pick, b is getting the yellow candy in the second pick.

So how do I find out? First is probability of A and the second is probability of B, given A has happened. Now what is the probability of A? The probability of A will be two divided by ten because there are two yellow candies and ten are the total candies. So this will be two divided by ten and multiplied by probability of B. And B is the second pick provided A has happened and provided A has happened, means that in the first pick we have already picked a yellow candy. So now in the plate what we are left with is one yellow candy and in total there are nine candies. So the second item here will not be two divided by ten, this will be one divided by nine. So this will be two divided by 90 or one by 45. So this was a case where we had dependent event happening of one affected the probability of the second event. So with this we complete the rule of Multiplication.

Now let’s move on to the Rule of Addition which is here. And this is also an important rule in probability. We use rule of multiplication when we have and when we are finding out the probability of and here instead of or if this was end, then we were using the rule of Multiplication. And when we are using or there we use the Rule of addition. Here we want to find out the probability of A or B. In the terms of Venn diagram, what we are looking here is not this central area, rather we are looking at this total whole area, either A or event B. And how do we find out? We find out this with this formula which is probability of A plus probability of B minus probability of both. Now, if this doesn’t make sense, let’s quickly look at this. So here I have the Venn diagram. Here I have a here I have B. So probability of A is the whole area of A, this is probability of A.

Now if I add a probability of B in this probability of B is this area. So if you add the probability of A and B, you get both these area area of A and B. But what you get is this area which is the intersection counted twice. So when you had probability of A, probability of B, this intersection area gets counted twice. So this is something which you need to remove and that will give you the whole area of A and B. And this is what we are doing here, minus the probability of both event A and B occurring. And when I say A and B, this means a intersection B. And this can be shown by this formula here. So PA union B, a union B is probability of A or B is equal to probability of A plus probability of B minus probability of A and B happening together.

Let’s take a simple example here. Let’s say our event A is getting one, two or three in a row of dice, either one or two or three, that’s event A. So let me put it here. A is one, two or three. Now event B is getting one, three or five, all the odd numbers. So B is one, three and five. So now if I ask you what is the probability of either event A or event B happening, that you can calculate using this formula. So the probability of A will be three by six, which is one by two. Probability of B will be again three by six, which is again one by two. And then we want to find out what is the probability of a intersection B. And here, if you look in the vein diagram in the intersection area, we have two items, one and three. So that means the probability of a intersection B will be two divided by six or one by three. So now if I have to find out probability of a union B or A or B, this will be one by two plus one by two.

8. IBM Baas Demo (DEMO ONLY) No support and not on the exam

Of a number and that should be the whole number and that should be the positive number. You cannot have a factorial of 1. 5, you cannot have factorial of minus two. So this has to be the positive number and this has to be the whole number. So the factorial of a number is given by that number multiplied by one less multiplied by one less till you reach one. The simplest example I have here is five. Factorial five factorial is equal to five multiplied by four multiplied by three multiplied by two multiplied by one. So if you multiply all these numbers that will be factorial five. But now if you are asking why do I need to learn about factorial five? Just wait for a few minutes and once we go to permutation and combination you will see the application of this factorial. I talked about factorial five. Now, factorial one is equal to one. That’s simple because one and then there you stop. Factorial zero is also equal to one. This is something which you need to keep in your mind that factorial zero is not zero.

Factorial zero is equal to one. So this was about factorial. Now coming to permutation and combinations, permutations and combinations are the way we can select number of items from a group. Let’s say you have five people and from them you want to select three people. How many ways you can select three people out of five? So let’s say if I give them number 12345, I can select one, two, three, I can select one, two, four, I can select one to five. So there are a number of ways I can select three people out of five. This is what permutation and combination will help us in finding out. Now, when I say permutation, permutation is the one in which the order is important. So when I say that I want to select three people from the team of five, then order matters. So whatever person I select first will become, let’s say the captain of the team. The second person which I select will become the vice captain and the third person which I select will become the treasurer.

Now here the order matters because whosoever gets selected first becomes the captain of the team, the second one becomes the vice captain. Whereas on the other hand, I just needed three people out of five irrespective of the order or the sequence that will be called as combination because in combination the order is not important. I just need three people out of these five. So whether I select the member number one as the first selection or as the second selection or as the third selection, that doesn’t make any difference. In case of combination we will look at example of each of these. Let’s start with permutation. Here is the simple example of permutation where repetition is allowed. So I was talking about permutation and combination and in each of these there is an option of repetition or repetition is not allowed. Let’s take the example of this lock.

This is a number lock which has numbers from zero to nine. So each wheel has ten numbers and there are four wheels here. Here repetition is allowed, that’s the first thing I know. I can have a number which is let’s say 2233, so I can repeat a number and why this is permutation, because here the order is important. The lock number is 1234. It’s not same as 4321 or 1324 because these are different permutations, these are different selections. So here the sequence or the order is important, that’s the reason it is permutation. So when you have a permutation and repetition is allowed, in that case you can use the simple formula which is n to the power R. N is the number of choices in each of these and R is that how many you are picking from this. So in this we have ten choices in each and we are picking four of these. The simplest way you can find this out is ten to the power four, which comes out to be 10,000, and which makes common sense as well, because in this lock there are 10,000 options available from 0004 zeros to nine. Nine nine. If you count these, these will be 10,000 choices, 10,000 ways you can have this number. So this was permutation with repetition at the bottom of this slide I have given Excel formula as well. So if you want to use Microsoft Excel and you put the formula as equal to permutation A and in the bracket ten comma four.

So you have ten choices and you are selecting four that will give you this number which is 10,000 coming to the next one, which is permutation without repetition. Now, let’s take an example. Here we have five players and from these five players I want to select three and the first selected will be the captain, the second will be the vice captain and the third selection will become the treasurer. How many ways I can select three people with order without repetition? So I cannot select one particular player two times to become the captain as well as the vice captain. So this is an example of permutation where order is important and repetition is not allowed. That you can find out with this simple formula which is NPR. And P here means permutation, which is equal to n factorial divided by n minus r factorial. And we learned about factorials just while ago and this is the application of using factorial. So here I’m looking at five p three. So there are five members and I want to select three of them which is equal to five factorial divided by five minus three. Factorial five factorial becomes five multiplied by four multiplied by three multiplied by two multiplied by one.

Five minus three, which is two factorial becomes two multiplied by one. So if I remove two multiplied by one. From each of these, this gives me five multiplied by four, multiplied by three is equal to 60. So there are 60 ways I can select three members out of this in a specific order. And for this, if you want to use Excel formula, you can use Excel formula as equal to permute five comma three in the bracket. In the previous example, when we were having repetition allowed there, we were using permutation A and we were giving these numbers. So this is permutation without repetition. Now, if this is permutation, let’s take the same example as combination. And as we earlier talked, in combination, the order is not important. So if order is not important, let’s say in the same example, what we want to do is we want to just select three players. We don’t want to assign them captain, vice captain or treasurer role, we just need three people from this.

So how many ways we can select? So here, in case of permutation, there were 60 ways. And if we have combination, let’s see, here is the formula for combination, combination without repetition. And of course here also we cannot have repetition, we cannot have player number one selected two times or three times, we just can select one player once. So now, if I want to select three players out of five, the formula for that is NCR and C. Here is combination. So earlier we were talking about NPR, where P was permutation. Here we are talking about C. C is the combination. And if you see the difference, here is this number here, which is R factorial. So there is additional R factorial in the denominator. So if I put all these values, this gives me five factorial divided by five minus three factorial and three factorial.

And if I solve this, this gives me number ten. Earlier, when we were talking of permutation, where order was important, the same number came as a 60. Here, when the order is not important, we just have ten ways to select three players out of five. Excel fabulous, for this is equal to combine combin five, comma three, and this will give you the value as ten. So rather than going through all these numbers n factorial, n minus R factorial, you can straight away put this into Excel and get the result as ten. So far we have talked about two cases of permutation and we have talked about one case of combination without repetition. And if repetition is allowed in combination, let’s look at that as well in the next slide, which is here. So here we have an example of combination. And once again, in combination, order does not matter. And here repetition is allowed, which was not allowed earlier. So what we have here as an example is that in a store there are five varieties of juice bottles. And if I go there and I want to buy three bottles because I have money for just three bottles, how many possible combinations I can buy. Maybe I can buy three bottles of a single variety or I might buy two of one variety and one of another variety, or I might just buy three different varieties because I want to buy three.

Only the formula for that is R plus n minus one factorial divided by r factorial n minus one factorial. So if you use this formula and this is the formula for with repetition, NCR, which is the previous formula, the formula which you used here is without repetition. So both these formula, NCR or NPR, both these formulas are for without repetition. So if I put values here, this gives me the value of 35. So there are 35 ways, 35 possible combinations of buying three bottles of juice out of five varieties which are available. And Excel formula for this is combine which is for the combination and A at the end basically represents that repetition is allowed. So this is with repetition and five comma three. This will give you a value of 35. So we have learned about four cases.

Let’s summarize these four cases on the next slide which is here. So here we have two cases of permutation where order is important and two cases of combination where order is not important. And here we have with repetition and without repetition. These are the formulas which we learned earlier, these four formulas and these at the bottom are Excel commands which you can put in Excel by putting equal to sign in front of that equal to permutation a and comma r or permute n comma r for without repetition, permutation.

And these are other two formulas for Excel. So this is the summary of all the formulas related to permutation and combination. We will be using many of these concepts as we move further and we talk about probability distributions. So once we have, let’s say binomial distribution there, we will be talking about these concepts related to NPR, NCR, etc. As we move further into this course.