Sample Space

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A Sample Space is a multiset that represents all possible outcomes of a random experiment.



For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).[2]
A well-defined sample space is one of three basic elements in a probabilistic model (a probability space); the other two are a well-defined set of possible events (a sigma-algebra) and a probability assigned to each event (a probability measure function).

In an experiment or any random trial, when we make a set of all the results or outcomes that are possible in that experiment or trial, that set is said to be a sample space of that particular experiment. For example, when we flip a coin there are only two possibilities; either to get a head or a tail. If we represent head by ‘H’ and tail be ‘T’ then the sample space of the experiment of flipping a coin, say S = {H, T}.
To find the probability of any event associated with a random experiment we divide the number of outcomes of that event with the total number of possible outcomes of the whole experiment that is the sample space.(...) In general sample space is denoted by three common symbols: U, S, Ω.
Commonly we used, ‘U’ to represents universal set. An experiment can have more than one sample space depending on the condition been set.


  • (Forbes et al., 2011) ⇒ Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). Statistical distributions. John Wiley & Sons.



  • (Hogg & Ledolter, 1987) ⇒ Robert V. Hogg and Johannes Ledolter. (1987). “Engineering Statistics. Macmillan Publishing Company.
    • Random experiments have outcomes that cannot be determined with certainty before the experiments are performed... The collection of all possible outcomes, namely [math]\displaystyle{ S }[/math] = {H,T}, is called the sample space. Suppose that we are interested in a subset [math]\displaystyle{ A }[/math] of our sample space; for example, in our case, let A={H} represent heads. Repeat this random experiment a number of times, say [math]\displaystyle{ n }[/math], and count the number of times, say [math]\displaystyle{ f }[/math], that the experiment ended in A. Here [math]\displaystyle{ f }[/math] is called the frequency of the event A and the ratio f/n is called the relative frequency of the event [math]\displaystyle{ A }[/math] in the [math]\displaystyle{ n }[/math] trials of the experiment.


  • (Larsen & Marx, 1986) ⇒ Richard J. Larsen, and Morris L. Marx. (1986). “An Introduction to Mathematical Statistics and Its Applications, 2nd edition." Prentice Hall
    • By an experiment we will mean any procedure that (1) can be repeated, theoretically, an infinite number of times; and (2) has a well-defined set of possible outcomes. Thus, rolling a pair of dice qualifies as an experiment; so does measuring a hypertensive's blood pressure or doing a stereographic analysis to determine the carbon content of moon rocks. Each of the potential eventualities of an experiment is referred to as a sample outcome, [math]\displaystyle{ s }[/math], and their totality is called the sample space, S. To signify the member of [math]\displaystyle{ s }[/math] in [math]\displaystyle{ S }[/math], we write sS. Any designated collection of sample outcomes, including individual outcomes, the entire sample space, and the null set, constitutes an event. The latter is said to occur if the outcome of the experiment is one of the members of that event.

  1. Albert, Jim (21 January 1998). "Listing All Possible Outcomes (The Sample Space)". Bowling Green State University. Retrieved June 25, 2013. 
  2. Larsen, R. J.; Marx, M. L. (2001). An Introduction to Mathematical Statistics and Its Applications (Third ed.). Upper Saddle River, NJ: Prentice Hall. p. 22. ISBN 9780139223037.