Simple Random Sample How many different simple random samples ...

Question

Simple Random Sample How many different simple random samples of size 7 can be obtained from a population whose size is 100?

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A library has 10 books available for donation. The library staff wants to select 3 different books from this collection to display on a special shelf. Additionally, they need to choose one of the displayed books to label as the book of the month. If the books are selected and labeled at random, what is the probability of selecting three specific books the staff had in mind for the display as well as 1 exact book for the book of the month label? Note. That the three specific books the staff had in mind for the display are predefined and are known in advance. Awesome. So it appears for this particular problem, we're ultimately trying to determine what the probability will be of selecting 3 specific books the staff had in mind for the display, as well as 1 exact book for the book of the month label. Awesome. So what is this probability value? That's what we're ultimately trying to solve for. So with that in mind, let's read off our multiple choice answers to see what our final answer might be, noting that they're all fractions. So A is 1 divided by 524, B is 1 divided by 360, C is 1 divided by 691, and D is 1 divided by 275. Awesome. So our first step that we need to take in order to solve this particular problem is we need to take a moment to understand what exactly we're trying to do. We need to understand our prompt. We need to note once again that we are tasked with the job of finding the probability of selecting 3 specific books from a collection of 10 total books. And then randomly selecting one of the displayed books to have the label as the label of book of the month. And this is a probability problem that involves both combinations for selecting the book and simple probability for selecting the book of the month. So now, our second step is we need to determine the total number of ways to select 3 books out of the 10 total options. So we need to start by calculating the total number of ways that we can select 3 books from the 10 total available books. Since the order in which we select the books doesn't matter, this is a combination problem, and because we're dealing with a combination problem, we need to recall and use the combination formula, and the combination formula states that C of NR. is equal to n factoral divided by R factoral multiplied by N minus R factoral. Fantastic. Where N is equal to 10 in this case, so let's make a note of this. So N is equal to 10, and this is the total number of books, and R in this case is going to be equal to 3, because this is the number of books that are gonna be selected. And now we need to substitute these values into our combination formula and solve. So that will mean that C of 10.3 will be equal to 10 factoral divided by 3. factoral multiplied by 10 minus 3 factoral, which when we plug that into our calculator, we will find that it's simply equal to 120. Fantastic. So now our next The point that we need to note is we need to note that since there are 120 different ways to select 3 books from 10, so ultimately, from what we've learned, that means that there's 120 different ways that 3 books can be selected from this 10 total amount of books. So there's 120 different combinations for 3 books being pulled away from the total of 10 books. So, with that said, Our next step now, our 3rd step is we need to figure out the number of favorable outcomes of selecting 3 specific books. So we need to note that there is only 1 way to select 3 specific books the staff had in mind because they are a fixed set. These three books are predefined and are known in advance. Therefore, as a result, the number of favorable outcomes is 1. So let's make a note of that. So favorable outcomes is just plain old number 1. So now, our 4th step that we need to consider is the total number of ways to label one of the 3 selected books as the book of the month. So once the 3 books are selected, there are 3 ways to label one of them as the book of the month, since there are 3 books to choose from. So we'll put a number 3 down as a note. Our 5th step now. we need to now at this point, calculate the probability, so we need to note note and recall that the total number of possible outcomes is the number of ways to select 3 books from 10, multiplied by the number of ways to select the book of the month from the 3 selected books. So that means that the total outcome is still called TO, so we're gonna call total outcomes TO, and the total outcomes is equal to 120, multiplied by 3. And if we multiply 120 by 3, we will find that 120 multiplied by 3 will be equal to 360. So the probability of selecting the three specific books from. So once again, so the probability of selecting the three specific books and correct book of the month is the ratio of favorable outcomes to the total number of possible outcomes, which will mean therefore, as a result, He so we could write this as P. Of so the capital P denotes the probability, so the probability of selecting three specific. Books And the correct book of the month, so book of month, I'll call it BOM for short. So, the possibility of, so the probability of selecting three specific books and the book of the month. Is going to be equal to 1 divided by 360, and that is our final answer. So know that I used abbreviations. So once again, the probability of selecting three specific books, so S3SB and the book of the month, so BOM is going to be equal to 1 divided by 360. And this corresponds with multiple choice answer letter B, which once again is 1 divided by 360. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Simple Random Sample How many different simple random samples of size 7 can be obtained from a population whose size is 100?

Key Concepts

Simple Random Sample

Combination Formula

Factorial Notation

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