Sample space

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File:Sample space.png
A visual representation of a finite sample space and events. The red oval is the event that a number is odd, and the blue oval is the event that a number is prime.

Template:Probability fundamentals

In probability theory, the sample space (also called sample description space,[1] possibility space,[2] or outcome space[3]) of an experiment or random trial is the set of all possible outcomes or results of that experiment.[4] A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points,[5] are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite.[6]

A subset of the sample space is an event, denoted by . If the outcome of an experiment is included in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} , then event Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} has occurred.[7]

For example, if the experiment is tossing a single coin, the sample space is the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{H,T\}} , where the outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} means that the coin is heads and the outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} means that the coin is tails.[8] The possible events are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=\{\}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=\{H\}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \{T\}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \{H,T\}} . For tossing two coins, the sample space is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{HH, HT, TH, TT\}} , where the outcome is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle HH} if both coins are heads, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle HT } if the first coin is heads and the second is tails, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle TH} if the first coin is tails and the second is heads, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle TT} if both coins are tails.[9] The event that at least one of the coins is heads is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \{HH,HT,TH\}} .

For tossing a single six-sided die one time, where the result of interest is the number of pips facing up, the sample space is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,2,3,4,5,6\}} .[10]

A well-defined, non-empty sample space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is one of three components in a probabilistic model (a probability space). The other two basic elements are a well-defined set of possible events (an event space), which is typically the power set of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is discrete or a σ-algebra on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} if it is continuous, and a probability assigned to each event (a probability measure function).[11]

A sample space can be represented visually by a rectangle, with the outcomes of the sample space denoted by points within the rectangle. The events may be represented by ovals, where the points enclosed within the oval make up the event.[12]

Conditions of a sample space

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A set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} with outcomes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_1, s_2, \ldots, s_n} (i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = \{s_1, s_2, \ldots, s_n\}} ) must meet some conditions in order to be a sample space:[13]

  • The outcomes must be mutually exclusive, i.e. if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_j} occurs, then no other Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} will take place, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall i,j=1,2,\ldots,n \quad i\neq j} .[6]
  • The outcomes must be collectively exhaustive, i.e. on every experiment (or random trial) there will always take place some outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i \in \Omega} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in \{1, 2, \ldots, n\}} .[6]
  • The sample space (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} ) must have the right granularity depending on what the experimenter is interested in. Irrelevant information must be removed from the sample space and the right abstraction must be chosen.

For instance, in the trial of tossing a coin, one possible sample space is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_1 = \{H,T\}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} is the outcome where the coin lands heads and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is for tails. Another possible sample space could be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_2 = \{(H,R), (H,NR), (T,R), (T,NR)\}} . Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} denotes a rainy day and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle NR} is a day where it is not raining. For most experiments, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_1} would be a better choice than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_2} , as an experimenter likely does not care about how the weather affects the coin toss.

Multiple sample spaces

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For many experiments, there may be more than one plausible sample space available, depending on what result is of interest to the experimenter. For example, when drawing a card from a standard deck of fifty-two playing cards, one possibility for the sample space could be the various ranks (Ace through King), while another could be the suits (clubs, diamonds, hearts, or spades).[4][14] A more complete description of outcomes, however, could specify both the denomination and the suit, and a sample space describing each individual card can be constructed as the Cartesian product of the two sample spaces noted above (this space would contain fifty-two equally likely outcomes). Still other sample spaces are possible, such as right-side up or upside down, if some cards have been flipped when shuffling.

Equally likely outcomes

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File:Coin tossing.JPG
Flipping a coin leads to a sample space composed of two outcomes that are almost equally likely.
A brass tack with point downward
Up or down? Flipping a brass tack leads to a sample space composed of two outcomes that are not equally likely.
Main article: Equally likely outcomes

Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely.[15] For any sample space with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} equally likely outcomes, each outcome is assigned the probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{N}} .[16] However, there are experiments that are not easily described by a sample space of equally likely outcomes—for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no physical symmetry to suggest that the two outcomes should be equally likely.[17]

Though most random phenomena do not have equally likely outcomes, it can be helpful to define a sample space in such a way that outcomes are at least approximately equally likely, since this condition significantly simplifies the computation of probabilities for events within the sample space. If each individual outcome occurs with the same probability, then the probability of any event becomes simply:[18]: 346–347 

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{P}(\text{event}) = \frac{\text{number of outcomes in event}}{\text{number of outcomes in sample space}}}

For example, if two fair six-sided dice are thrown to generate two uniformly distributed integers, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_2} , each in the range from 1 to 6, inclusive, the 36 possible ordered pairs of outcomes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (D_1,D_2)} constitute a sample space of equally likely events. In this case, the above formula applies, such as calculating the probability of a particular sum of the two rolls in an outcome. The probability of the event that the sum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1 + D_2} is five is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4}{36}} , since four of the thirty-six equally likely pairs of outcomes sum to five.

If the sample space was all of the possible sums obtained from rolling two six-sided dice, the above formula can still be applied because the dice rolls are fair, but the number of outcomes in a given event will vary. A sum of two can occur with the outcome Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(1,1)\}} , so the probability is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{36}} . For a sum of seven, the outcomes in the event are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{(1,6), (6,1), (2,5), (5,2), (3,4),(4,3)\}} , so the probability is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{6}{36}} .[19]

Simple random sample

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Main article: Simple random sample

In statistics, inferences are made about characteristics of a population by studying a sample of that population's individuals. In order to arrive at a sample that presents an unbiased estimate of the true characteristics of the population, statisticians often seek to study a simple random sample—that is, a sample in which every individual in the population is equally likely to be included.[18]: 274–275  The result of this is that every possible combination of individuals who could be chosen for the sample has an equal chance to be the sample that is selected (that is, the space of simple random samples of a given size from a given population is composed of equally likely outcomes).[20]

Infinitely large sample spaces

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In an elementary approach to probability, any subset of the sample space is usually called an event.[9] However, this gives rise to problems when the sample space is continuous, so that a more precise definition of an event is necessary. Under this definition only measurable subsets of the sample space, constituting a σ-algebra over the sample space itself, are considered events.

An example of an infinitely large sample space is measuring the lifetime of a light bulb. The corresponding sample space would be [0, ∞).[9]

See also

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References

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  1. ↑ Stark, Henry; Woods, John W. (2002). Probability and Random Processes with Applications to Signal Processing (3rd ed.). Pearson. p. 7. ISBN 9788177583564.
  2. ↑ Forbes, Catherine; Evans, Merran; Hastings, Nicholas; Peacock, Brian (2011). Statistical Distributions (4th ed.). Wiley. p. 3. ISBN 9780470390634.
  3. ↑ Hogg, Robert; Tannis, Elliot; Zimmerman, Dale (December 24, 2013). Probability and Statistical Inference. Pearson Education, Inc. p. 10. ISBN 978-0321923271. The collection of all possible outcomes... is called the outcome space.
  4. ↑ 4.0 4.1 Albert, Jim (1998-01-21). "Listing All Possible Outcomes (The Sample Space)". Bowling Green State University. Archived from the original on 2019-09-13. Retrieved 2013-06-25.
  5. ↑ Soong, T. T. (2004). Fundamentals of probability and statistics for engineers. Chichester: Wiley. ISBN 0-470-86815-5. OCLC 55135988.
  6. ↑ 6.0 6.1 6.2 "UOR_2.1". web.mit.edu. Retrieved 2019-11-21.
  7. ↑ Ross, Sheldon (2010). A First Course in Probability (PDF) (8th ed.). Pearson Prentice Hall. p. 23. ISBN 978-0136033134. Archived from the original (PDF) on 2021-07-24. Retrieved 2021-12-02.
  8. ↑ Dekking, F.M. (Frederik Michel), 1946- (2005). A modern introduction to probability and statistics : understanding why and how. Springer. ISBN 1-85233-896-2. OCLC 783259968.CS1 maint: multiple names: authors list (link)
  9. ↑ 9.0 9.1 9.2 "Sample Space, Events and Probability" (PDF). Mathematics at Illinois. Archived from the original (PDF) on 2023-05-23. Retrieved 2019-11-21.
  10. ↑ Larsen, R. J.; Marx, M. L. (2001). An Introduction to Mathematical Statistics and Its Applications (3rd ed.). Upper Saddle River, NJ: Prentice Hall. p. 22. ISBN 9780139223037.
  11. ↑ LaValle, Steven M. (2006). Planning Algorithms (PDF). Cambridge University Press. p. 442.
  12. ↑ "Sample Spaces, Events, and Their Probabilities". saylordotorg.github.io. Retrieved 2019-11-21.
  13. ↑ Tsitsiklis, John (Spring 2018). "Sample Spaces". Massachusetts Institute of Technology. Retrieved July 9, 2018.
  14. ↑ Jones, James (1996). "Stats: Introduction to Probability - Sample Spaces". Richland Community College. Retrieved 2013-11-30.
  15. ↑ Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher's Edition (Classics ed.). Prentice Hall. p. 633. ISBN 0-13-165711-9.
  16. ↑ "Equally Likely outcomes" (PDF). University of Notre Dame.
  17. ↑ "Chapter 3: Probability" (PDF). Coconino Community College.
  18. ↑ 18.0 18.1 Yates, Daniel S.; Moore, David S.; Starnes, Daren S. (2003). The Practice of Statistics (2nd ed.). New York: Freeman. ISBN 978-0-7167-4773-4. Archived from the original on 2005-02-09.
  19. ↑ "Probability: Rolling Two Dice". www.math.hawaii.edu. Retrieved 2021-12-17.
  20. ↑ "Simple Random Samples". web.ma.utexas.edu. Retrieved 2019-11-21.
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