Probabilità/Introduzione: differenze tra le versioni

<math>p_{A} = \lim_{n\to \infty}\frac{n_{A}}{n}</math>
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It is of course impossible to conduct an infinite number of trials. However, it usually suffices to conduct a large number of trials, where the standard of large depends on the probability being measured and how accurate a measurement we need.
 
A note on this definition of probability: How do we know the sequence <math>\frac{n_{A}}{n}</math> in the limit will converge to the same result every time, or that it will converge at all? The unfortunate answer is that we don't. To see this, consider an experiment consisting of flipping a coin an infinite number of times. We are interested in the probability of heads coming up. Imagine the result is the following sequence:
:''HTHHTTHHHHTTTTHHHHHHHHTTTTTTTTHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTT''...
with each run of <math>k</math> heads and <math>k</math> tails being followed by another run twice as long. For this example, the sequence <math>\frac{n_{A}}{n}</math> oscillates between roughly <math>\frac{1}{3}</math> and <math>\frac{2}{3}</math> and doesn't converge.
 
We might expect such sequences to be unlikely, and we would be right. It will be shown later that the probability of such a run is 0, as is a sequence that converges to anything other than the underlying probability of the event. However, such examples make it clear that the limit in the definition above does not express convergence in the more familiar sense, but rather some kind of convergence in probability. The problem of formulating exactly what this means belongs to axiomatic probability theory.
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Immaginiamo che il risultato sia l seguente sequenza:
:''HTHHTTHHHHTTTTHHHHHHHHTTTTTTTTHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTT''...
dove per ogni <math>k</math> risultati testa e <math>k</math> risultati croce segue un numero doppio di risultati. Seguendo questo esempio il rapporto <math>\frac{n_{A}}{n}</math> oscilla tra circa <math>\frac{1}{3}</math> e <math>\frac{2}{3}</math> senza convergere nel rapporto <math>\frac{1}{2}</math>.
dove ogni
 
 
Potremmo aspettarci che tali sequenze di risultati siano improbabili, e avremmo ragione. Come mostrato dopo la probabilità di una striscia di risultati come questa è pari a 0, come una serie di risultati che converge verso la probabilità statistica dell'evento.
It is of course impossible to conduct an infinite number of trials. However, it usually suffices to conduct a large number of trials, where the standard of large depends on the probability being measured and how accurate a measurement we need.
Il limite di questo tipo di probabilità sta proprio nel definire chiaramente come determinare la convergenza di un tale evento verso la probabilità statistica, compito che spetta alla teoria della probabilità assiomatica.
 
A note on this definition of probability: How do we know the sequence <math>\frac{n_{A}}{n}</math> in the limit will converge to the same result every time, or that it will converge at all? The unfortunate answer is that we don't. To see this, consider an experiment consisting of flipping a coin an infinite number of times. We are interested in the probability of heads coming up. Imagine the result is the following sequence:
:''HTHHTTHHHHTTTTHHHHHHHHTTTTTTTTHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTT''...
with each run of <math>k</math> heads and <math>k</math> tails being followed by another run twice as long. For this example, the sequence <math>\frac{n_{A}}{n}</math> oscillates between roughly <math>\frac{1}{3}</math> and <math>\frac{2}{3}</math> and doesn't converge.
 
We might expect such sequences to be unlikely, and we would be right. It will be shown later that the probability of such a run is 0, as is a sequence that converges to anything other than the underlying probability of the event. However, such examples make it clear that the limit in the definition above does not express convergence in the more familiar sense, but rather some kind of convergence in probability. The problem of formulating exactly what this means belongs to axiomatic probability theory.
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===Teoria assiomatica della probabilità===
Utente anonimo