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April 14th, 2018, 08:55 AM   #1
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Question Question - Sample space in probability

I would like to know how to solve the following question:
Throw a cube until you get the number 6, then stop throwing.
a) What is the sample space of the experiment?
b) Let's call the event to throw the cube n times En. How much points from the sample space are within En?
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April 14th, 2018, 09:38 AM   #2
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The English on this question is somewhat fractured.

The experiment consists of throwing a standard die until it comes up 6.

$E_n$ is the event that the experiment is finished in n throws.

The only question that makes sense to me is "what is the sample space?"
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April 14th, 2018, 10:22 AM   #3
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And the only problem with "The only question that makes sense to me is 'what is the sample space?'" is that there is no such thing as the sample space. What you should us as a sample space depends on what questions you are looking to answer and/or how easy it is to determine probabilities for each.

While there are some who will call my definitions a little loose, and others too restrictive (especially the last one):
  • I'm going to assume that you cube is a standard six-sided die, with the numbers "1" thru "6" printed on the sides. You really should have said that.
  • A random experiment is some activity where the results are uncertain. But you haven't made it clear what you consider to be "one" experiment, although you seem to imply that it is the set of rolls ending in a 6. But you could have called each roll a separate experiment.
  • A random variable is any way you can assign values, usually numbers, to any possible result of what you would call the entire experiment. So the total number of 1s is a random variable. The result of the the third roll, including whether it occurs, is a random variable. The sum of all the rolls is a random variable.
  • Once you define your random variables, an outcome is a description of the result of a single experiment, in terms of them. So "There were five rolls, including two 1s and a 3 on the third roll, for a sum of 15" is an outcome if you use all of the random variables I described. If you use only the total number rolls, the same outcome (and many others) are all included in the outcome "five rolls."
  • The sample space you want is the set of all possible outcomes that could occur, as described by your random variables. That means every single combination that gives a values to each one.
  • An event is something you don't seem to need for this question. It means any combination of your outcomes. So "At least three rolls, but less than ten rolls" is an event. If your only random variable is the number of rolls, this event can be expressed as "the number of rolls is in {3,4,5,6,7,8,9}". But many times people will use "outcome" and "event" interchangeably.

So, nobody can tell you what sample space you should use, until you specify what random variables you consider important. And "how many points" is not a question you can ask about a sample space. It is a set that has elements, not points, and what they are is subjective.
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April 16th, 2018, 03:54 AM   #4
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Quote:
Originally Posted by lola19991 View Post
I would like to know how to solve the following question:
Throw a cube until you get the number 6, then stop throwing.
a) What is the sample space of the experiment?
The "sample space" is the set or all strings of integers, between 1 and 5, with "6" on the end.

Quote:
b) Let's call the event to throw the cube n times En. How much points from the sample space are within En?
Since you threw the cube n times and there must be a 6 at the end but no 6 before that, we can have all strings of n-1 digits from 1 to 5. There are 5 possible integers for the first number and the same 5 for the second number so $\displaystyle 5(5)= 5^2$ two digit strings, then the same 5 for the third number so $\displaystyle 5^3$ for three digits, etc. Continuing that way, there will be $\displaystyle 5^{n-1}$ strings of n-1 digits from 1 to 5, followed by a "6".
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April 16th, 2018, 05:28 AM   #5
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Quote:
Originally Posted by Country Boy View Post
The "sample space" is the set or all strings of integers, between 1 and 5, with "6" on the end.
But this is the point, which is often neglected by many textbooks: there is no such thing as THE sample space. Multiple people analyzing the same experiment may very well come up with very different sample spaces. For this experiment, there are multiple sample spaces possible, some more elaborate and some less elaborate. I agree your sample space is the ideal one, but it should be clear many other choices are possible.
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