View source for Bernoulli trials

[[Category:Probability]]

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Independent trials, each one of which can have only two results ( "success"  or  "failure" ) such that the probabilities of the results do not change from one trial to another. Bernoulli trials are one of the principal schemes considered in probability theory.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156901.png" /> be the probability of success, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156902.png" /> be the probability of failure, and let 1 denote the occurrence of success, while 0 denotes the occurrence of a failure. The probability of a given sequence of successful or unsuccessful events, e.g.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156903.png" /></td> </tr></table>

is equal to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156904.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156905.png" /> is the number of successful events in the series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156906.png" /> trials under consideration. Many frequently occurring probability distributions are connected with Bernoulli trials. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156907.png" /> be the random variable which is equal to the number of successes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156908.png" /> Bernoulli trials. The probability of the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b0156909.png" /> is then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569010.png" /></td> </tr></table>

i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569011.png" /> has a [[Binomial distribution|binomial distribution]]. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569012.png" />, this distribution can be approximated by the [[Normal distribution|normal distribution]] or by the [[Poisson distribution|Poisson distribution]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569013.png" /> be the number of trials prior to the first success. The probability of the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569014.png" /> then is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569015.png" /></td> </tr></table>

i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569016.png" /> has a [[Geometric distribution|geometric distribution]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569017.png" /> is the number of failures which precede the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569018.png" />-th appearance of a successful result, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569019.png" /> has the so-called [[Negative binomial distribution|negative binomial distribution]]. The number of successful outcomes of Bernoulli trials can be represented as the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569020.png" /> of independent random variables, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569021.png" /> if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569022.png" />-th trial was a success, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569023.png" /> otherwise. This is why many important laws of probability theory dealing with sums of independent variables were originally established for Bernoulli trial schemes (cf. [[Bernoulli theorem|Bernoulli theorem]] ((weak) [[Law of large numbers|Law of large numbers]]); [[Strong law of large numbers|Strong law of large numbers]]; [[Law of the iterated logarithm|Law of the iterated logarithm]]; [[Central limit theorem|Central limit theorem]]; etc.).

A rigorous study of infinite sequences of Bernoulli trials requires the introduction of a [[Probability measure|probability measure]] in the space of infinite sequences of zeros and ones. This may be done directly or by the method illustrated for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569024.png" /> below. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569025.png" /> be a number, uniformly distributed on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569026.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569028.png" /> or 1, be the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569029.png" /> into a binary fraction. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569031.png" /> are independent and assume the values 0 and 1 with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569032.png" /> each, i.e. the succession of zeros and ones in the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569033.png" /> is described by the Bernoulli trial scheme with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569034.png" />. However, the measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569035.png" /> can also be specified so as to obtain Bernoulli trials with any desired <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569036.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015690/b01569037.png" /> the measure obtained is singular with respect to the Lebesgue measure).

Bernoulli trials are often treated geometrically (cf. [[Bernoulli random walk|Bernoulli random walk]]). Certain probabilities of a large number of events connected with Bernoulli trials were computed in the initial stage of development of probability theory in the context of the ruin problem.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Gnedenko,   "The theory of probability" , Chelsea, reprint  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Feller,   "An introduction to probability theory and its applications" , '''1–2''' , Wiley  (1957–1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Kac,   "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer.  (1963)</TD></TR></table>