WEBVTT mathematics/statistics/son
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Hi and welcome to www.educator.com.
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We are going to be talking about the addition rule for disjoints events.
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First, we wanted to go over again what disjoint events mean?
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and then we will talk briefly about the meaning of the word or because it has a slightly different meaning in statistics than in regular life.
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Then we will talk about how to calculate the probability of a or b and that method is going to be the addition rule.
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We are going to talk about the addition rule when A and B are disjoint events, as well as when they are not disjoint events.
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Let us talk about disjointed events.
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remember we have said before disjoint events are mutually exclusive and they cannot both happen at the same time.
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so you cannot draw one card from a deck and have it be both a Jack and Ace.
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It has to either be a Jack or Ace, you cannot have it be both at the same time.
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Where can you select a student at random from high school and get both the junior and senior.
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We have to be just one or the other.
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However, you can select a card from a deck and have that it be either a jack or a heart.
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they are not mutually exclusive.
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They can both happen at the same time.
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They do not always have to but they can.
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if we wanted to draw this idea as a picture we may show these two events this is the sample space of all the possible things in the world that might happen.
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Here is when A is true, your card is a Jack.
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Here is when B is true, your card is an Ace.
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Notice that there is no overlap between those two things.
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There is no part of this space both the Jack and Ace parts are true.
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In this picture we show a non disjoint events.
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Here we might have as the space for all the events when jack is true but here we have the space where the card is a heart.
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Here we have this space where it is both a jack and a heart.
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Here we see that it is possible to be both jack and heart.
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Here we might have jacks and hearts like jack of cloves or spades.
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Here we have hearts that are not jacks, the king of hearts or the ace of hearts.
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Now let us talk about the meaning of or.
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In regular life, we usually use the word or like this, would you like soup or salad?
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Would you like it to be red or blue?
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Typically we mean would you like one of these things or the other one?
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You cannot have both at the same time.
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You can only choose one.
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If you say you like this you mean to say you have to pay for the other one.
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How do in math, statistics, and computer science, we are not talking about choices.
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We are talking about truth values.
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A or B means that either A is true or B is true, or they are both true A and B.
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Basically the big idea of this is that at least one of these is true.
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If both are true then you fit the world, at least one of them is true.
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In that way or means slightly different in statistics lingo.
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Now we could talk about how to calculate the probability of A or B and that is called the addition rule.
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First let us talk about the addition rule for disjoint events.
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The way you could think about this is what is the probability that you will land in this space or this space?
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What is it asking you what is the probability that you will land in one of this shaded areas out of all the other possibilities?
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In that way you might see that you might add together these 2 probabilities.
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If A and B are disjoint, we need to first talk about what is the probability of A and B is true?
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You could see that there are no points here.
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There are no space, they are both A and B are covered in.
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That overall space between A and B and because of that we would say 0.
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There are 0 part of this space where both A and B is true.
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We only have A being true or B being true but not both.
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Here we have to calculate what is the probability of A or B.
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You could easily see, you might want to add together probability of A and B.
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This is 25% and this is 25%, maybe we have 50% likely having A or B.
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That only works for disjoint events.
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Here if it is disjoint events the world is simple.
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If we only have one disjoint events, let us imply that same logic.
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Remember we have talked about disjoint events we are asking what is the probability that you are lined within these shaded event spaces.
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P(a or b) means what is the probability in one of these shaded spaces.
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It is the same idea in non disjoint spaces.
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I want you to watch carefully.
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It is what the probability landing in A or in B.
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Notice that this space is counted twice.
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When you say p(a or b), what you want to know is this 8 shaped area and we do not want to count that part twice.
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How can we come up with a general addition rule that will work for either non disjoint events.
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You might want to start off thinking about this in the same that we did before.
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First it might be helpful to add together the probability of A and B but we have counted this part twice.
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It will be helpful to take one of those out.
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We might subtract the probability of A and B.
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In that way we can use one of these out and I will get just this area.
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The reason why this is called the general addition rule is because it actually works for disjoints events as well.
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Let us try for disjoint events.
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We have probability of A or B, and we have probability of A + B, we will subtract the probability of A and B.
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Here what is the probability of A and B?
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It is 0 here.
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What we are doing is adding these probabilities and subtracting 0.
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That means the addition rule for disjoint events.
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This general addition rule works for all events, both disjoint and non disjoint.
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The general addition rule is important and very useful method of contact.
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There is a different way you could write it.
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I just want to briefly show that to you here.
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I’m just writing down the generalize addition rule and rewrite this using slightly different notation.
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Many of the frequent notation you might see is the use of union and intersection.
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Union this goes to or.
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When 2 countries unite, you are thinking about 2 pieces coming together and now both pieces count.
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That is the picture that you should think about or.
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When Alaska is united to the rest of United States together.
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It is or whether here or here.
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The other piece of notation you need to know is the intersection notation.
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This one match perfectly to and.
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The idea is where do they intersect?
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Where A intersect with B?
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We could just rewrite this using this notation.
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This is often used for sets.
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You could just use this notation.
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Instead of or we can use union A and union B and that would mean the probability of A and B.
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We do not have to change anything there - the probability of A intersection B.
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This is just saying this whole area shaded in = p(a + b) and take out this little section.
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Let us move on to some examples.
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Let us say that I have chosen a person at random, which of these are mutually exclusive?
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By using mutually exclusive you want to think of the same term as disjoint events.
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They will be used interchangeably.
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Has ridden a roller coaster or Ferris wheel, does one prevent the other from happening?
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There are people who have both ridden roller coasters and Ferris wheel so I would say that these are non disjoint or not mutually exclusive.
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There are nothing that prevents people from doing both.
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I think I can have both songs on iTunes.
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Has brown eyes has brown hair?
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Here too you could have both brown eyes and brown hair or you could have just brown eyes but other colored hair.
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You could just have brown hair and other colored eyes.
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This is also non disjoint.
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Is it left handed or right handed?
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Your dominant hand is just one or the other.
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I would say this is probably disjoint.
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What pops into my mind right now are the ambidextrous people but even them they prefer one hand versus the other.
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That is just disjoint for now.
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Has had chicken pox never had chicken pox?
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This one is definitely disjoint because we cannot have both have it and not at the same time.
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Suppose we flip a coin 3 times what is the probability that you will have a combination of 1 head and 2 tails.
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You have to remember the lessons from last time as well as the addition rule.
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First we have to figure out what the entire sample space looks like.
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The whole square would be, for example if we toss the coin 3 times that would mean that is 2 different power of 8.
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First flip and second flip, third flip.
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Half of these 8 are supposed to be heads and half of them are tails.
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I am just going to start off with that.
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Second one, half of these have to be heads and half have to be tails.
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Half of these have to be heads and half of these have to be tails.
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Same thing here.
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I do not see the pattern but it is just cuts in half every single time.
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The last one half of 2 alternates.
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Now we know that each of these are probability of 1/8.
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We can use the addition rule to figure out what is the probability that you have a combination of 1 heads and 2 tails.
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We could just find other one heads and 2 tails and we add those probabilities together because these 3 things are disjoint events.
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You cannot have both heads-tails-tails, and tails-head-tails at the same time nor can you have tails-tails-head at the same time either.
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We have the probability of 1 heads and 2 tails = 3/8.
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Ann and Bill are planning their engagement party, the room will hold 200 people so they agreed that Ann will invite 100 friends and Bill will invite 100 friends.
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Everyone invited in the party showed up but only 140 people turned up, what must have happened?
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It might be helpful to think about this as a picture.
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Here all the people that Ann invited and here all the people that Bill invited, and together it is going to be 140 people.
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There are some portions of difference that must have had overlapped.
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We are trying to figure that out.
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Right now we are looking at frequency instead of percentages, but obviously you can turn these to percentages if you want.
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I’m just going to keep it in frequency form.
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We are looking for the probability of A or B – the probability of A and B.
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You can turn this probability statement into the total number of people.
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We are just going to keep it as this.
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We have 100 of Ann’s friends, 100 of Bill’s friends, and we do not know the overlap.
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We do know that 140 people showed up eventually.
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We do know that part.
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The way you could do this is to divide everything by 200 because that is the total number of invited.
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If we do this and we have 200 – p( a and b) I want to add this to both sides so that I can make it positive.
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I am going to do that right here, 200 – 140 and that is going to be 60.
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The probability of A and B are the frequency in this case of A and B is 60.
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That makes sense, 60 people are here them how many people did Ann know? That is 30 people.
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The same with Bill because he has invited 100 people and 60 of them knows Ann.
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When you add all of these up, it makes 140.
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In a community 80% of households in car insurance or homeowner insurance, 30% carry homeowner’s insurance and 50% in car insurance.
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If the household is picked at random what is the probability that they are both has an insurance?
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Once again we could use the general addition rule to figure this out.
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Because we do not know the probability that the household owns both but we do know the probability that they either own car or home.
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They at least own one and that is 80%.
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40% are just home owners insurance and 50% owns just car insurance.
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We want to know what is the probability that they both have insurance.
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I will write the general version, it does not matter whether p of c or h comes first – the probability of C and H.
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I will put in the numbers here.
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I will subtract these from both sides and that gave me 10%.
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10% both have insurance.
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That is for the addition rule.
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