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A family consisting of three persons $-A, B,$ and $C-$ belongs to a medical clinic that always has a doctor at each of stations $1,2,$ and $3 .$ During a certain week, each mer of the family visits the clinic once and is assigned at random to a station. The experiment consists of recording the station number for each member. One outcome is $(1,2,1)$ for $A$ to station $1, B$ to station $2,$ and $C$ to station $1 .$

(a) List the 27 outcomes in the sample space.

(b) List all outcomes in the event that all three members go to the same station.

(c) List all outcomes in the event that all members go to different stations.

(d) List all outcomes in the event that no one goes to station 2 .

(a) {111, 112, 113, 121, 122, 123, 131, 132, 133, 211, 212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313,

321, 322, 323, 331, 332, 333} $\\$

(b) {111, 222, 333} $\\$

(c) {123, 132, 213, 231, 312, 321} $\\$

(d) {111, 113, 131, 133, 311, 313, 331, 333}

Probability Topics

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Okay. In this case, there are three family members E b and C. Okay, there are three family members that belong to the medical clinic, and, uh, there are three stations. Okay. And on each station, there is always a doctor. Now what happens is during a certain week, each member of the family visits the clinic and is assigned to a random station. So what we're doing is we're recording the station number for each of the member, right? And in the question, there is an example given to us. For example, One comma, two commas one. Now, if we see this set off numbers, it means that they went to station aid. Sorry, even to station one B went to station to and see again. Went to station one. Okay, Now, what do they want us to do? They want us to list the sample space. Okay, First of all, there are three stations and there are three people. So how many different arrangements are possible for them? Okay, let's say that there are three stations. It ain't so. How many different days are possible? Three into three into three. Three multiplied by three, multiplied by three. Which means what? Which is nothing but 27. So which means there are total 27 possible outcomes right now. What is the first question? That is more to question A question says list or 27 outcomes in the samples piece. Okay, so we have to list all the 27 outcomes we saw here. The answer is indeed 27. Now, how do we list all of them? Let us begin. There are pairs off three numbers, right? 12 and three. So how many different combinations are possible? Traditional? So we just have to list all of them. So let's say I start with this one. 11 When? When? To 11 three. Okay. Then comes. One, 21 12 to then comes 12 three. Similarly, one, 31 one, three do One, three, three. Okay, now let's begin with two. This is too 11 This is 21 to this is 213 Then comes 2 to 1, then comes, Do do do. Then comes Do do three. Then comes to 31 232 23 three. Now similarly, we will also again begin with three 311 312 31 three. Here comes 321 Then comes 32 to 323 Then comes 331 three, 32 On three, 33 These are the 27 possible outcomes that form our sample space. Okay, this was our part. A what is part B list? All outcomes in the event that all three members go to the same station, that is all three numbers are the same. How many outcomes are there in this case? I think there will only be three outcomes. Right. If these are the three stations, all three of them go here like a B and C. All three go here or all three of them go here or all three of them go here. Right. So these are going to be my possible outcomes. Okay, So now what do I write? How will I write this? This is going to be 111 or to To To or three, three, three. Okay. These are going to be my answers. I think there's also packed. See, Patsy is list all outcomes in the event that all members go to different stations. All of them go to different stations. Okay, so how many possible outcomes out there? Let's say that there are these three stations in the first one. Any three Anyone of the three can group in the second one. Any two. And over here, anyone. So what is the total number off outcomes? In this case, it is going to be three into 21 which is nothing but six. Right. So what I'm doing is I'm arranging 12 and three. Okay. Now I am arranging 12 entries. So this is going to be one comma. Two commas. Three. This is one possible arrangement. One comma, three com a tube. Right? Or I can do this with two comma, three comma, one or two comma, one comma. Three. Okay. And then let's say he goes to the third station. Three comma, comma one and then three comma, one comma to see all of them, go to different stations. Now, what is part B list? All outcomes where no one goes to station to. Okay, so we do not want station to right. If I don't want station to, which means that this is fact de. Okay, this is a This is B. This is C. There are these three different people. Okay, Now all of them have only two options. Either station one or station three. So this guy has two options. He has two options, and he also has two options to enter. To enter to is nothing but eight. So there are going to be eight possible outcomes in this case. So let's begin. Let's start with one comma. Okay? No one goes to station to write one comma, one comma one or this is one comma, one comma three. Okay, this is two comma, one comma one, or this is two comma, one comma. Three. Oh, excuse me. Sorry. We are not going to station to, right? We are not going to station to. Okay, so this is going to be just on. This is going to be three. This is going to be three. Okay, so we have right. And when more possibility is three comma, three comma one. And then there is also three comma. Three com. A tree. Okay, there is one comma, three comma. One. What else can be there? We are missing one more. So this is mhm. One comma, one comma, 31 comma one comma 11 comma, three comma one. And there has to be one more. Just a moment. There is 113 have written. Yes. Then 111131 And 13311131 And 133133 is also possible. So I think they should also be there. Yes, I have missed this 133 So these other possible outcomes in part deep and these are going to be our answers.