39:
1819:
1807:
3033:
1067:
2373:—that is, a sample in which every individual in the population is equally likely to be included. 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).
1890:
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
1938:
2118:
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
1074:
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.
1797:
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.
2047:
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
1793:(clubs, diamonds, hearts, or spades). 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
1784:
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
1897:
1677:
2310:
1356:
1231:
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1396:
2337:
2182:
2113:
954:
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887:
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1774:
1747:
2155:
686:
1469:
1101:
648:
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515:
1887:
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.
584:
1999:
1972:
1293:
1266:
2389:. 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
1070:
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.
1720:
834:
811:
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1854:
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977:
555:
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480:
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1836:
Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. For any sample space with
330:
2696:
2700:
1883:. 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
2921:
2821:
2793:
1933:{\displaystyle \mathrm {P} ({\text{event}})={\frac {\text{number of outcomes in event}}{\text{number of outcomes in sample space}}}}
3053:
2396:
An example of an infinitely large sample space is measuring the lifetime of a light bulb. The corresponding sample space would be
1475:
depending on what the experimenter is interested in. Irrelevant information must be removed from the sample space and the right
1571:
323:
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2187:
1298:
1165:
2647:
2521:
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2753:
2672:
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316:
304:
263:
1789:, one possibility for the sample space could be the various ranks (Ace through King), while another could be the
364:
194:
130:
2393:
subsets of the sample space, constituting a σ-algebra over the sample space itself, are considered events.
2946:
1106:
242:
103:
3037:
1944:
1891:
each individual outcome occurs with the same probability, then the probability of any event becomes simply:
1401:
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2429:
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984:
419:
98:
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903:
691:
214:
1859:
839:
2004:
273:
268:
157:
142:
2051:
1831:
1752:
1725:
252:
123:
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2866:
399:"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be
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2605:
2358:
2122:
653:
396:
147:
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2719:
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288:
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152:
118:
3004:
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65:
8:
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237:
179:
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816:
793:
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1839:
1776:, as an experimenter likely does not care about how the weather affects the coin toss.
1682:
1551:
1531:
1365:, i.e. on every experiment (or random trial) there will always take place some outcome
1038:
1014:
994:
962:
836:
if both coins are tails. The event that at least one of the coins is heads is given by
540:
520:
465:
445:
425:
344:
224:
113:
53:
30:
2001:, each in the range from 1 to 6, inclusive, the 36 possible ordered pairs of outcomes
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2414:
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980:
485:
For example, if the experiment is tossing a single coin, the sample space is the set
372:
283:
189:
88:
2665:
A modern introduction to probability and statistics : understanding why and how
2369:
of the true characteristics of the population, statisticians often seek to study a
108:
38:
2365:
of that population's individuals. In order to arrive at a sample that presents an
2817:
2409:
408:
368:
184:
135:
897:
199:
3047:
2872:
2745:
2682:
2479:
Forbes, Catherine; Evans, Merran; Hastings, Nicholas; Peacock, Brian (2011).
2115:, since four of the thirty-six equally likely pairs of outcomes sum to five.
404:
72:
2586:
16:
Set of all possible outcomes or results of a statistical trial or experiment
1790:
1786:
1483:
For instance, in the trial of tossing a coin, one possible sample space is
380:
299:
209:
2434:
1032:
2566:
2382:
1476:
1056:
219:
60:
48:
2868:
Algebra and
Trigonometry: Functions and Applications, Teacher's Edition
2528:
The collection of all possible outcomes... is called the outcome space.
2456:
Probability and Random
Processes with Applications to Signal Processing
2354:
1884:
400:
379:
or results of that experiment. A sample space is usually denoted using
77:
23:
2662:
2770:
988:
383:, and the possible ordered outcomes, or sample points, are listed as
2966:
2512:
Hogg, Robert; Tannis, Elliot; Zimmerman, Dale (December 24, 2013).
1943:
For example, if two fair six-sided dice are thrown to generate two
983:). The other two basic elements are a well-defined set of possible
204:
1856:
equally likely outcomes, each outcome is assigned the probability
1818:
387:
in the set. It is common to refer to a sample space by the labels
3032:
2742:
An
Introduction to Mathematical Statistics and Its Applications
1806:
1066:
415:
2945:
Yates, Daniel S.; Moore, David S.; Starnes, Daren S. (2003).
1233:) must meet some conditions in order to be a sample space:
893:
2478:
1672:{\displaystyle \Omega _{2}=\{(H,R),(H,NR),(T,R),(T,NR)\}}
2568:
Fundamentals of probability and statistics for engineers
1814:
composed of two outcomes that are almost equally likely.
1722:
is a day where it is not raining. For most experiments,
896:
one time, where the result of interest is the number of
813:
if the first coin is tails and the second is heads, and
2305:{\displaystyle \{(1,6),(6,1),(2,5),(5,2),(3,4),(4,3)\}}
979:
is one of three components in a probabilistic model (a
2385:, any subset of the sample space is usually called an
1351:{\displaystyle \forall i,j=1,2,\ldots ,n\quad i\neq j}
557:
means that the coin is tails. The possible events are
2318:
2190:
2163:
2125:
2094:
2054:
2007:
1980:
1953:
1900:
1862:
1842:
1826:
composed of two outcomes that are not equally likely.
1755:
1728:
1705:
1685:
1574:
1568:
is for tails. Another possible sample space could be
1554:
1534:
1489:
1457:
1404:
1371:
1301:
1274:
1247:
1226:{\displaystyle \Omega =\{s_{1},s_{2},\ldots ,s_{n}\}}
1168:
1109:
1089:
1041:
1017:
997:
965:
906:
842:
819:
796:
773:
750:
694:
656:
624:
592:
563:
543:
523:
491:
468:
448:
428:
2845:"Stats: Introduction to Probability - Sample Spaces"
2184:. For a sum of seven, the outcomes in the event are
790:
if the first coin is heads and the second is tails,
2511:
2545:"Listing All Possible Outcomes (The Sample Space)"
2331:
2304:
2176:
2149:
2107:
2080:
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1993:
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1154:
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881:
828:
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736:
680:
642:
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578:
549:
529:
509:
474:
454:
434:
2944:
2376:
2357:, inferences are made about characteristics of a
442:. If the outcome of an experiment is included in
3045:
2794:"Sample Spaces, Events, and Their Probabilities"
2663:Dekking, F.M. (Frederik Michel), 1946- (2005).
1078:
1548:is the outcome where the coin lands heads and
1822:Up or down? Flipping a brass tack leads to a
688:. For tossing two coins, the sample space is
537:means that the coin is heads and the outcome
324:
2695:: CS1 maint: multiple names: authors list (
2299:
2191:
2144:
2126:
1666:
1588:
1515:
1503:
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731:
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631:
605:
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573:
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504:
492:
2816:
2810:
2739:
2699:) CS1 maint: numeric names: authors list (
2453:
1801:
331:
317:
1779:
1155:{\displaystyle s_{1},s_{2},\ldots ,s_{n}}
987:(an event space), which is typically the
2940:
2938:
2864:
2744:(3rd ed.). Upper Saddle River, NJ:
2538:
2536:
1817:
1805:
1065:
2824:. Massachusetts Institute of Technology
2768:
2342:
959:A well-defined, non-empty sample space
3046:
2720:"Sample Space, Events and Probability"
2542:
2516:. Pearson Education, Inc. p. 10.
1441:{\displaystyle i\in \{1,2,\ldots ,n\}}
2935:
2842:
2836:
2564:
2533:
2514:Probability and Statistical Inference
2454:Stark, Henry; Woods, John W. (2002).
2714:
2712:
2710:
2628:
2600:
2598:
2596:
2458:(3rd ed.). Pearson. p. 7.
2740:Larsen, R. J.; Marx, M. L. (2001).
1521:{\displaystyle \Omega _{1}=\{H,T\}}
13:
1926:number of outcomes in sample space
1902:
1757:
1730:
1576:
1491:
1458:
1385:
1302:
1169:
1090:
14:
3070:
3025:
2707:
2593:
2558:
3031:
2547:. Bowling Green State University
1391:{\displaystyle s_{i}\in \Omega }
37:
3054:Experiment (probability theory)
2997:
2981:"Probability: Rolling Two Dice"
2973:
2914:
2893:
2858:
2786:
2762:
2733:
2487:(4th ed.). Wiley. p.
2332:{\displaystyle {\frac {6}{36}}}
2177:{\displaystyle {\frac {1}{36}}}
2108:{\displaystyle {\frac {4}{36}}}
1338:
949:{\displaystyle \{1,2,3,4,5,6\}}
900:facing up, the sample space is
892:For tossing a single six-sided
737:{\displaystyle \{HH,HT,TH,TT\}}
2656:
2622:
2505:
2472:
2447:
2377:Infinitely large sample spaces
2296:
2284:
2278:
2266:
2260:
2248:
2242:
2230:
2224:
2212:
2206:
2194:
2141:
2129:
2034:
2008:
1914:
1906:
1876:{\displaystyle {\frac {1}{N}}}
1749:would be a better choice than
1663:
1648:
1642:
1630:
1624:
1609:
1603:
1591:
882:{\displaystyle E=\{HH,HT,TH\}}
104:Collectively exhaustive events
1:
2632:A First Course in Probability
2440:
2381:In an elementary approach to
2040:{\displaystyle (D_{1},D_{2})}
2847:. Richland Community College
1079:Conditions of a sample space
7:
2769:LaValle, Steven M. (2006).
2403:
2081:{\displaystyle D_{1}+D_{2}}
1923:number of outcomes in event
1810:Flipping a coin leads to a
1769:{\displaystyle \Omega _{2}}
1742:{\displaystyle \Omega _{1}}
1055:if it is continuous, and a
10:
3075:
2951:(2nd ed.). New York:
2948:The Practice of Statistics
2929:Coconino Community College
2865:Foerster, Paul A. (2006).
2780:Cambridge University Press
2543:Albert, Jim (1998-01-21).
2430:Event (probability theory)
2346:
1829:
1059:assigned to each event (a
418:of the sample space is an
2901:"Equally Likely outcomes"
2483:Statistical Distributions
2150:{\displaystyle \{(1,1)\}}
767:if both coins are heads,
681:{\displaystyle E=\{H,T\}}
2922:"Chapter 3: Probability"
2908:University of Notre Dame
2312:, so the probability is
2157:, so the probability is
1699:denotes a rainy day and
353:sample description space
274:Law of total probability
269:Conditional independence
158:Exponential distribution
143:Probability distribution
3005:"Simple Random Samples"
2727:Mathematics at Illinois
1832:Equally likely outcomes
1802:Equally likely outcomes
1464:{\displaystyle \Omega }
1363:collectively exhaustive
1096:{\displaystyle \Omega }
744:, where the outcome is
643:{\displaystyle E=\{T\}}
611:{\displaystyle E=\{H\}}
510:{\displaystyle \{H,T\}}
253:Conditional probability
2798:saylordotorg.github.io
2629:Ross, Sheldon (2010).
2333:
2306:
2178:
2151:
2109:
2082:
2041:
1995:
1968:
1934:
1877:
1850:
1827:
1815:
1780:Multiple sample spaces
1770:
1743:
1716:
1693:
1673:
1562:
1542:
1522:
1465:
1442:
1392:
1352:
1289:
1268:occurs, then no other
1262:
1227:
1156:
1097:
1071:
1049:
1025:
1005:
973:
950:
883:
830:
807:
784:
761:
738:
682:
644:
612:
580:
579:{\displaystyle E=\{\}}
551:
531:
511:
476:
456:
436:
195:Continuous or discrete
148:Bernoulli distribution
2871:(Classics ed.).
2843:Jones, James (1996).
2640:Pearson Prentice Hall
2571:. Chichester: Wiley.
2565:Soong, T. T. (2004).
2334:
2307:
2179:
2152:
2110:
2083:
2042:
1996:
1994:{\displaystyle D_{2}}
1969:
1967:{\displaystyle D_{1}}
1945:uniformly distributed
1935:
1878:
1851:
1821:
1809:
1771:
1744:
1717:
1694:
1674:
1563:
1543:
1523:
1466:
1443:
1393:
1361:The outcomes must be
1353:
1290:
1288:{\displaystyle s_{i}}
1263:
1261:{\displaystyle s_{j}}
1237:The outcomes must be
1228:
1157:
1098:
1069:
1050:
1026:
1006:
974:
951:
884:
831:
808:
785:
762:
739:
683:
645:
613:
581:
552:
532:
512:
477:
457:
437:
153:Binomial distribution
3040:at Wikimedia Commons
2371:simple random sample
2349:Simple random sample
2343:Simple random sample
2316:
2188:
2161:
2123:
2092:
2052:
2005:
1978:
1951:
1898:
1860:
1840:
1753:
1726:
1703:
1683:
1572:
1552:
1532:
1487:
1455:
1402:
1369:
1299:
1272:
1245:
1166:
1107:
1087:
1039:
1015:
995:
963:
904:
840:
817:
794:
771:
748:
692:
654:
622:
590:
561:
541:
521:
517:, where the outcome
489:
466:
446:
426:
409:uncountably infinite
279:Law of large numbers
248:Marginal probability
173:Poisson distribution
22:Part of a series on
3059:Space (mathematics)
2985:www.math.hawaii.edu
2772:Planning Algorithms
2420:Space (mathematics)
1061:probability measure
238:Complementary event
180:Probability measure
168:Pareto distribution
163:Normal distribution
2329:
2302:
2174:
2147:
2105:
2078:
2037:
1991:
1964:
1930:
1873:
1846:
1828:
1816:
1766:
1739:
1715:{\displaystyle NR}
1712:
1689:
1669:
1558:
1538:
1518:
1461:
1451:The sample space (
1438:
1388:
1348:
1285:
1258:
1239:mutually exclusive
1223:
1152:
1093:
1072:
1045:
1021:
1001:
969:
946:
879:
829:{\displaystyle TT}
826:
806:{\displaystyle TH}
803:
783:{\displaystyle HT}
780:
760:{\displaystyle HH}
757:
734:
678:
640:
608:
576:
547:
527:
507:
472:
452:
432:
345:probability theory
289:Boole's inequality
225:Stochastic process
114:Mutual exclusivity
31:Probability theory
3036:Media related to
3009:web.ma.utexas.edu
2962:978-0-7167-4773-4
2425:Set (mathematics)
2415:Probability space
2367:unbiased estimate
2327:
2172:
2103:
1928:
1927:
1924:
1912:
1871:
1849:{\displaystyle N}
1795:Cartesian product
1692:{\displaystyle R}
1561:{\displaystyle T}
1541:{\displaystyle H}
1473:right granularity
1295:will take place,
1048:{\displaystyle S}
1031:is discrete or a
1024:{\displaystyle S}
1004:{\displaystyle S}
981:probability space
972:{\displaystyle S}
550:{\displaystyle T}
530:{\displaystyle H}
475:{\displaystyle E}
455:{\displaystyle E}
435:{\displaystyle E}
357:possibility space
341:
340:
243:Joint probability
190:Bernoulli process
89:Probability space
3066:
3035:
3019:
3018:
3016:
3015:
3001:
2995:
2994:
2992:
2991:
2977:
2971:
2970:
2965:. Archived from
2942:
2933:
2932:
2926:
2918:
2912:
2911:
2905:
2897:
2891:
2890:
2862:
2856:
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2834:
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2831:
2829:
2818:Tsitsiklis, John
2814:
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2790:
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2724:
2716:
2705:
2704:
2694:
2686:
2660:
2654:
2653:
2638:(8th ed.).
2637:
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2020:
2019:
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1998:
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1970:
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1962:
1939:
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1936:
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1913:
1910:
1905:
1882:
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1879:
1874:
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1864:
1855:
1853:
1852:
1847:
1775:
1773:
1772:
1767:
1765:
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1737:
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1713:
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1678:
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1584:
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1567:
1565:
1564:
1559:
1547:
1545:
1544:
1539:
1527:
1525:
1524:
1519:
1499:
1498:
1471:) must have the
1470:
1468:
1467:
1462:
1447:
1445:
1444:
1439:
1397:
1395:
1394:
1389:
1381:
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1199:
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1153:
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1150:
1132:
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1119:
1118:
1102:
1100:
1099:
1094:
1054:
1052:
1051:
1046:
1030:
1028:
1027:
1022:
1010:
1008:
1007:
1002:
978:
976:
975:
970:
955:
953:
952:
947:
888:
886:
885:
880:
835:
833:
832:
827:
812:
810:
809:
804:
789:
787:
786:
781:
766:
764:
763:
758:
743:
741:
740:
735:
687:
685:
684:
679:
649:
647:
646:
641:
617:
615:
614:
609:
585:
583:
582:
577:
556:
554:
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548:
536:
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528:
516:
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513:
508:
481:
479:
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473:
461:
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453:
441:
439:
438:
433:
375:of all possible
333:
326:
319:
109:Elementary event
41:
19:
18:
3074:
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3069:
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2898:
2894:
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2859:
2850:
2848:
2841:
2837:
2827:
2825:
2822:"Sample Spaces"
2820:(Spring 2018).
2815:
2811:
2802:
2800:
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2791:
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2767:
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2756:
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2708:
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2534:
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2510:
2506:
2499:
2477:
2473:
2466:
2452:
2448:
2443:
2410:Parameter space
2406:
2397:
2379:
2351:
2345:
2319:
2317:
2314:
2313:
2189:
2186:
2185:
2164:
2162:
2159:
2158:
2124:
2121:
2120:
2095:
2093:
2090:
2089:
2072:
2068:
2059:
2055:
2053:
2050:
2049:
2028:
2024:
2015:
2011:
2006:
2003:
2002:
1985:
1981:
1979:
1976:
1975:
1958:
1954:
1952:
1949:
1948:
1920:
1909:
1901:
1899:
1896:
1895:
1863:
1861:
1858:
1857:
1841:
1838:
1837:
1834:
1804:
1782:
1760:
1756:
1754:
1751:
1750:
1733:
1729:
1727:
1724:
1723:
1704:
1701:
1700:
1684:
1681:
1680:
1579:
1575:
1573:
1570:
1569:
1553:
1550:
1549:
1533:
1530:
1529:
1494:
1490:
1488:
1485:
1484:
1479:must be chosen.
1456:
1453:
1452:
1403:
1400:
1399:
1376:
1372:
1370:
1367:
1366:
1300:
1297:
1296:
1279:
1275:
1273:
1270:
1269:
1252:
1248:
1246:
1243:
1242:
1214:
1210:
1195:
1191:
1182:
1178:
1167:
1164:
1163:
1146:
1142:
1127:
1123:
1114:
1110:
1108:
1105:
1104:
1088:
1085:
1084:
1081:
1040:
1037:
1036:
1016:
1013:
1012:
996:
993:
992:
964:
961:
960:
905:
902:
901:
841:
838:
837:
818:
815:
814:
795:
792:
791:
772:
769:
768:
749:
746:
745:
693:
690:
689:
655:
652:
651:
623:
620:
619:
591:
588:
587:
562:
559:
558:
542:
539:
538:
522:
519:
518:
490:
487:
486:
467:
464:
463:
447:
444:
443:
427:
424:
423:
337:
185:Random variable
136:Bernoulli trial
17:
12:
11:
5:
3072:
3062:
3061:
3056:
3042:
3041:
3027:
3026:External links
3024:
3021:
3020:
2996:
2972:
2969:on 2005-02-09.
2961:
2934:
2913:
2892:
2885:
2857:
2835:
2809:
2785:
2782:. p. 442.
2761:
2754:
2748:. p. 22.
2732:
2706:
2673:
2655:
2649:978-0136033134
2648:
2642:. p. 23.
2621:
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2557:
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2523:978-0321923271
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2378:
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2361:by studying a
2347:Main article:
2344:
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2099:
2075:
2071:
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2018:
2014:
2010:
1988:
1984:
1961:
1957:
1941:
1940:
1919:
1916:
1908:
1904:
1870:
1867:
1845:
1830:Main article:
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1130:
1126:
1122:
1117:
1113:
1103:with outcomes
1092:
1080:
1077:
1044:
1033:σ-algebra
1020:
1000:
968:
945:
942:
939:
936:
933:
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927:
924:
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822:
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799:
779:
776:
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753:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
677:
674:
671:
668:
665:
662:
659:
639:
636:
633:
630:
627:
607:
604:
601:
598:
595:
575:
572:
569:
566:
546:
526:
506:
503:
500:
497:
494:
482:has occurred.
471:
451:
431:
339:
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336:
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328:
321:
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294:
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292:
291:
286:
284:Bayes' theorem
281:
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258:
257:
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232:
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215:Observed value
212:
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200:Expected value
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27:
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3010:
3006:
3000:
2986:
2982:
2976:
2968:
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2958:
2954:
2950:
2949:
2941:
2939:
2930:
2923:
2917:
2909:
2902:
2896:
2888:
2886:0-13-165711-9
2882:
2878:
2874:
2873:Prentice Hall
2870:
2869:
2861:
2846:
2839:
2823:
2819:
2813:
2799:
2795:
2789:
2781:
2774:
2773:
2765:
2757:
2755:9780139223037
2751:
2747:
2746:Prentice Hall
2743:
2736:
2728:
2721:
2715:
2713:
2711:
2702:
2698:
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2680:
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2674:1-85233-896-2
2670:
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2641:
2634:
2633:
2625:
2611:
2607:
2601:
2599:
2597:
2588:
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2578:0-470-86815-5
2574:
2570:
2569:
2561:
2546:
2539:
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2529:
2525:
2519:
2515:
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2500:
2498:9780470390634
2494:
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2465:9788177583564
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2016:
2012:
1986:
1982:
1959:
1955:
1946:
1917:
1894:
1893:
1892:
1888:
1886:
1868:
1865:
1843:
1833:
1825:
1820:
1813:
1808:
1799:
1796:
1792:
1788:
1787:playing cards
1777:
1761:
1734:
1709:
1706:
1686:
1660:
1657:
1654:
1651:
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1633:
1627:
1621:
1618:
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1606:
1600:
1597:
1594:
1585:
1580:
1555:
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1512:
1509:
1506:
1500:
1495:
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1474:
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1234:
1215:
1211:
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1201:
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843:
823:
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800:
797:
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754:
751:
728:
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722:
719:
716:
713:
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707:
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672:
669:
666:
660:
657:
634:
628:
625:
602:
596:
593:
567:
564:
544:
524:
501:
498:
495:
483:
469:
462:, then event
449:
429:
422:, denoted by
421:
417:
412:
410:
407:infinite, or
406:
402:
398:
397:universal set
394:
390:
386:
382:
378:
374:
370:
366:
362:
361:outcome space
358:
354:
351:(also called
350:
346:
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327:
322:
320:
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134:
133:
132:
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117:
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112:
110:
107:
105:
102:
101:
100:
97:
95:
92:
90:
87:
86:
85:
84:
79:
76:
74:
73:Indeterminism
71:
67:
64:
63:
62:
59:
55:
52:
51:
50:
47:
46:
45:
44:
40:
36:
35:
32:
29:
28:
25:
21:
20:
3038:Sample space
3012:. Retrieved
3008:
2999:
2988:. Retrieved
2984:
2975:
2967:the original
2947:
2928:
2916:
2907:
2895:
2867:
2860:
2849:. Retrieved
2838:
2826:. Retrieved
2812:
2801:. Retrieved
2797:
2788:
2771:
2764:
2741:
2735:
2726:
2667:. Springer.
2664:
2658:
2631:
2624:
2613:. Retrieved
2609:
2567:
2560:
2549:. Retrieved
2527:
2513:
2507:
2482:
2474:
2455:
2449:
2395:
2380:
2352:
2117:
1942:
1889:
1835:
1824:sample space
1823:
1812:sample space
1811:
1783:
1482:
1472:
1362:
1238:
1082:
1073:
958:
891:
484:
413:
392:
388:
381:set notation
360:
356:
352:
349:sample space
348:
342:
305:Tree diagram
300:Venn diagram
264:Independence
210:Markov chain
94:Sample space
93:
2610:web.mit.edu
2383:probability
2088:is five is
1477:abstraction
1063:function).
1057:probability
220:Random walk
61:Determinism
49:Probability
3048:Categories
3014:2019-11-21
2990:2021-12-17
2875:. p.
2851:2013-11-30
2803:2019-11-21
2615:2019-11-21
2551:2013-06-25
2441:References
2398:[0, ∞)
2391:measurable
2359:population
2355:statistics
1947:integers,
1885:thumb tack
1241:, i.e. if
367:or random
365:experiment
131:Experiment
78:Randomness
24:statistics
2691:cite book
2683:783259968
2606:"UOR_2.1"
2435:σ-algebra
1758:Ω
1731:Ω
1577:Ω
1492:Ω
1459:Ω
1427:…
1409:∈
1386:Ω
1383:∈
1343:≠
1330:…
1303:∀
1205:…
1170:Ω
1137:…
1091:Ω
989:power set
405:countably
124:Singleton
2587:55135988
2404:See also
1679:. Here,
1528:, where
391:, Ω, or
385:elements
377:outcomes
363:) of an
205:Variance
2953:Freeman
2828:July 9,
371:is the
119:Outcome
2959:
2883:
2752:
2681:
2671:
2646:
2585:
2575:
2520:
2495:
2462:
2363:sample
1162:(i.e.
1083:A set
985:events
650:, and
416:subset
401:finite
395:(for "
347:, the
66:System
54:Axioms
2925:(PDF)
2904:(PDF)
2776:(PDF)
2723:(PDF)
2636:(PDF)
2387:event
1911:event
1791:suits
420:event
369:trial
359:, or
99:Event
2957:ISBN
2881:ISBN
2830:2018
2750:ISBN
2701:link
2697:link
2679:OCLC
2669:ISBN
2644:ISBN
2583:OCLC
2573:ISBN
2518:ISBN
2493:ISBN
2460:ISBN
1974:and
1398:for
898:pips
2877:633
2353:In
1035:on
1011:if
991:of
894:die
373:set
343:In
3050::
3007:.
2983:.
2955:.
2937:^
2927:.
2906:.
2879:.
2796:.
2778:.
2725:.
2709:^
2693:}}
2689:{{
2677:.
2608:.
2595:^
2581:.
2535:^
2526:.
2491:.
2400:.
2339:.
2325:36
2170:36
2101:36
956:.
889:.
618:,
586:,
414:A
411:.
403:,
355:,
3017:.
2993:.
2931:.
2910:.
2889:.
2854:.
2832:.
2806:.
2758:.
2729:.
2703:)
2685:.
2652:.
2618:.
2589:.
2554:.
2501:.
2489:3
2468:.
2322:6
2300:}
2297:)
2294:3
2291:,
2288:4
2285:(
2282:,
2279:)
2276:4
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2270:3
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2261:)
2258:2
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2252:5
2249:(
2246:,
2243:)
2240:5
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2234:2
2231:(
2228:,
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2222:1
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2216:6
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2210:,
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2204:6
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2198:1
2195:(
2192:{
2167:1
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2139:1
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2130:(
2127:{
2098:4
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2070:D
2066:+
2061:1
2057:D
2035:)
2030:2
2026:D
2022:,
2017:1
2013:D
2009:(
1987:2
1983:D
1960:1
1956:D
1918:=
1915:)
1907:(
1903:P
1869:N
1866:1
1844:N
1762:2
1735:1
1710:R
1707:N
1687:R
1667:}
1664:)
1661:R
1658:N
1655:,
1652:T
1649:(
1646:,
1643:)
1640:R
1637:,
1634:T
1631:(
1628:,
1625:)
1622:R
1619:N
1616:,
1613:H
1610:(
1607:,
1604:)
1601:R
1598:,
1595:H
1592:(
1589:{
1586:=
1581:2
1556:T
1536:H
1516:}
1513:T
1510:,
1507:H
1504:{
1501:=
1496:1
1448:.
1436:}
1433:n
1430:,
1424:,
1421:2
1418:,
1415:1
1412:{
1406:i
1378:i
1374:s
1358:.
1346:j
1340:i
1336:n
1333:,
1327:,
1324:2
1321:,
1318:1
1315:=
1312:j
1309:,
1306:i
1281:i
1277:s
1254:j
1250:s
1221:}
1216:n
1212:s
1208:,
1202:,
1197:2
1193:s
1189:,
1184:1
1180:s
1176:{
1173:=
1148:n
1144:s
1140:,
1134:,
1129:2
1125:s
1121:,
1116:1
1112:s
1043:S
1019:S
999:S
967:S
944:}
941:6
938:,
935:5
932:,
929:4
926:,
923:3
920:,
917:2
914:,
911:1
908:{
877:}
874:H
871:T
868:,
865:T
862:H
859:,
856:H
853:H
850:{
847:=
844:E
824:T
821:T
801:H
798:T
778:T
775:H
755:H
752:H
732:}
729:T
726:T
723:,
720:H
717:T
714:,
711:T
708:H
705:,
702:H
699:H
696:{
676:}
673:T
670:,
667:H
664:{
661:=
658:E
638:}
635:T
632:{
629:=
626:E
606:}
603:H
600:{
597:=
594:E
574:}
571:{
568:=
565:E
545:T
525:H
505:}
502:T
499:,
496:H
493:{
470:E
450:E
430:E
393:U
389:S
332:e
325:t
318:v
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