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to be hit. Since the square has area 1, the probability that the dart will hit any particular subregion of the square is equal to the area of that subregion. For example, the probability that the dart will hit the right half of the square is 0.5, since the right half has area 0.5.
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In this case, any infinite sequence of heads and tails is a possible outcome of the experiment. However, any particular infinite sequence of heads and tails has probability 0 of being the exact outcome of the (infinite) experiment. This is because the
1342:" will also happen almost surely (i.e., with probability 1). But if instead of an infinite number of flips, flipping stops after some finite time, say 1,000,000 flips, then the probability of getting an all-heads sequence,
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1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of
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In general, an event can happen "almost surely", even if the probability space in question includes outcomes which do not belong to the event—as the following examples illustrate.
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of the unit square. Since the area of the diagonals of the square is 0, the probability that the dart will land exactly on a diagonal is 0. That is, the dart will
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not land on a diagonal), even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point.
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to be strictly between 0 and 1. In fact, the same result even holds in non-standard analysis—where infinitesimal probabilities are allowed.
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by assumption. The result is the same no matter how much we bias the coin towards heads, so long as we constrain
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1933:(January 2006). "A Sharp Threshold for Random Graphs with a Monochromatic Triangle in Every Edge Coloring".
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if a tail is flipped. For this particular coin, it is assumed that the probability of flipping a head is
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Consider the case where a (possibly biased) coin is tossed, corresponding to the probability space
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and the assumption that each flip's outcome is independent of all the others (i.e., they are
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Now, suppose an experiment were conducted where the coin is tossed repeatedly, with outcomes
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Some examples of the use of this concept include the strong and uniform versions of the
1824:; Marx, Maarten; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007).
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with a non-zero probability for each outcome, there is no difference between
97:); however, this distinction becomes important when the sample space is an
41:"Probability 1" redirects here. For Rudolf Carnap's notion of "probability
1737:, which says that probabilities should almost never be set as zero or one
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1986:. Algorithms and Combinatorics. Vol. 22. Springer. p. 4.
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assumption implies that the probability of flipping all heads over
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Next, consider the event that the dart hits exactly a point in the
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Moreover, the event "the sequence of tosses contains at least one
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The notion of almost sureness depends on the probability measure
2033:. Cambridge Mathematical Textbooks. Cambridge University Press.
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in the square, in such a way that each point in the square is
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93:(since having a probability of 1 entails including all the
2014:. Vol. 1: Foundations. Cambridge University Press.
30:"Almost never" redirects here. For the Biota album, see
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1892:"How probable is an infinite sequence of heads?"
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2012:Diffusions, Markov Processes, and Martingales
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2010:Rogers, L. C. G.; Williams, David (2000).
1749:, a theorem using the aforementioned terms
1016:{\displaystyle X_{i}(\omega )=\omega _{i}}
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716:{\displaystyle (\{H,T\},2^{\{H,T\}},P)}
1820:Grädel, Erich; Kolaitis, Phokion G.;
7:
1254:{\displaystyle n\rightarrow \infty }
216:{\displaystyle E\in {\mathcal {F}}}
1983:The Strange Logic of Random Graphs
1890:Williamson, Timothy (2007-07-01).
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367:{\displaystyle E\subseteq \Omega }
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108:, the continuity of the paths of
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1581:) is true a.a.s. when, for some
580:{\displaystyle \left(\!P\right)}
1726:, for "almost sure convergence"
1724:Convergence of random variables
1561:vertices with edge probability
1419:{\displaystyle 1-p^{1,000,000}}
821:{\displaystyle P(H)=p\in (0,1)}
1797:"Almost surely - Math Central"
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2031:Probability with Martingales
1440:asymptotically almost surely
1430:Asymptotically almost surely
1941:(845). AMS Bookstore: 3–4.
1753:List of mathematical jargon
1054:records the outcome of the
398:) happens almost surely if
47:Probability interpretations
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1978:"0. Two Starting Examples"
1683:, this is referred to as "
1493:{\displaystyle G(n,p_{n})}
1444:convergence in probability
1292:{\displaystyle p\in (0,1)}
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348:. More generally, any set
341:{\displaystyle P(E^{C})=0}
128:describes the opposite of
64:(sometimes abbreviated as
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1976:Spencer, Joel H. (2001).
644:Tossing a coin repeatedly
2063:Mathematical terminology
2029:Williams, David (1991).
862:{\displaystyle P(T)=1-p}
36:Almost Never (TV series)
1908:10.1093/analys/67.3.173
1747:Infinite monkey theorem
1741:Degenerate distribution
650:Infinite monkey theorem
114:infinite monkey theorem
1861:Probability Essentials
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604:Imagine throwing a
2058:Probability theory
1705:Mathematics portal
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1718:Almost everywhere
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1574:{\displaystyle p}
1554:{\displaystyle n}
1458:, the statement "
1335:{\displaystyle T}
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1095:{\displaystyle n}
1067:{\displaystyle i}
544:{\displaystyle E}
524:{\displaystyle P}
442:{\displaystyle N}
302:not occurring is
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140:Formal definition
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16:(Redirected from
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2052:Categories
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1806:2019-11-16
1782:2019-11-16
1685:almost all
1235:. Letting
648:See also:
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