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{{MSC|60-01}}
  
 
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.
 
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.
+
Let $  p $
 +
be the probability of success, let $  q = 1 - p $
 +
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>
+
$$
 +
1 \ \  0 \ \  0 \ \  1 \ \  1 \ \  0 \ \  1 \ \  0 \  \dots \  1,
 +
$$
  
 
is equal to
 
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>
+
$$
 +
p \  q \  q \  p \  p \  q \  p \  q \  \dots \  p \  = \
 +
p  ^ {m} q ^ {n - m} ,
 +
$$
 +
 
 +
where  $  m $
 +
is the number of successful events in the series of  $  n $
 +
trials under consideration. Many frequently occurring probability distributions are connected with Bernoulli trials. Let  $  S _ {n} $
 +
be the random variable which is equal to the number of successes in  $  n $
 +
Bernoulli trials. The probability of the event  $  \{ S _ {n} = k \} $
 +
is then
  
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
+
$$
 +
\left (
 +
{n \atop k}
 +
\right )
 +
p  ^ {k} q  ^ {n-k} ,\ \
 +
k = 0 \dots n,
 +
$$
  
<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.  $  S _ {n} $
 +
has a [[Binomial distribution|binomial distribution]]. As  $  n \rightarrow \infty $,
 +
this distribution can be approximated by the [[Normal distribution|normal distribution]] or by the [[Poisson distribution|Poisson distribution]]. Let  $  Y _ {1} $
 +
be the number of trials prior to the first success. The probability of the event  $  \{ Y _ {1} = k \} $
 +
then is
  
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
+
$$
 +
q  ^ {k} p ,\ \
 +
k = 0,\  1 \dots
 +
$$
  
<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.  $  Y _ {1} $
 +
has a [[Geometric distribution|geometric distribution]]. If  $  Y _ {r} $
 +
is the number of failures which precede the  $  r $-
 +
th appearance of a successful result,  $  Y _ {r} $
 +
has the so-called [[Negative binomial distribution|negative binomial distribution]]. The number of successful outcomes of Bernoulli trials can be represented as the sum  $  X _ {1} + \dots + X _ {n} $
 +
of independent random variables, in which  $  X _ {j} = 1 $
 +
if the  $  j $-
 +
th trial was a success, and  $  X _ {j} = 0 $
 +
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.).
  
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  $  p = q = 1/2 $
 +
below. Let  $  \omega $
 +
be a number, uniformly distributed on the segment  $  (0,\  1) $,
 +
and let
  
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
+
$$
 +
\omega \  = \  \sum _ {j=1} ^ \infty
  
<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>
+
\frac{X _ {j} ( \omega )}{2 ^ j}
 +
,
 +
$$
  
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).
+
where $  X _ {j} ( \omega ) = 0 $
 +
or 1, be the expansion of $  \omega $
 +
into a binary fraction. Then the $  X _ {j} $,
 +
$  j = 1,\  2 \dots $
 +
are independent and assume the values 0 and 1 with probability $  1/2 $
 +
each, i.e. the succession of zeros and ones in the expansion of $  \omega $
 +
is described by the Bernoulli trial scheme with $  p = 1/2 $.  
 +
However, the measure on $  (0,\  1) $
 +
can also be specified so as to obtain Bernoulli trials with any desired $  p $(
 +
if $  p \neq 1/2 $
 +
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.
 
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.

Latest revision as of 13:12, 6 February 2020


2010 Mathematics Subject Classification: Primary: 60-01 [MSN][ZBL]

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 $ p $ be the probability of success, let $ q = 1 - p $ 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.

$$ 1 \ \ 0 \ \ 0 \ \ 1 \ \ 1 \ \ 0 \ \ 1 \ \ 0 \ \dots \ 1, $$

is equal to

$$ p \ q \ q \ p \ p \ q \ p \ q \ \dots \ p \ = \ p ^ {m} q ^ {n - m} , $$

where $ m $ is the number of successful events in the series of $ n $ trials under consideration. Many frequently occurring probability distributions are connected with Bernoulli trials. Let $ S _ {n} $ be the random variable which is equal to the number of successes in $ n $ Bernoulli trials. The probability of the event $ \{ S _ {n} = k \} $ is then

$$ \left ( {n \atop k} \right ) p ^ {k} q ^ {n-k} ,\ \ k = 0 \dots n, $$

i.e. $ S _ {n} $ has a binomial distribution. As $ n \rightarrow \infty $, this distribution can be approximated by the normal distribution or by the Poisson distribution. Let $ Y _ {1} $ be the number of trials prior to the first success. The probability of the event $ \{ Y _ {1} = k \} $ then is

$$ q ^ {k} p ,\ \ k = 0,\ 1 \dots $$

i.e. $ Y _ {1} $ has a geometric distribution. If $ Y _ {r} $ is the number of failures which precede the $ r $- th appearance of a successful result, $ Y _ {r} $ has the so-called negative binomial distribution. The number of successful outcomes of Bernoulli trials can be represented as the sum $ X _ {1} + \dots + X _ {n} $ of independent random variables, in which $ X _ {j} = 1 $ if the $ j $- th trial was a success, and $ X _ {j} = 0 $ 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 ((weak) Law of large numbers); Strong law of large numbers; Law of the iterated logarithm; Central limit theorem; etc.).

A rigorous study of infinite sequences of Bernoulli trials requires the introduction of a 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 $ p = q = 1/2 $ below. Let $ \omega $ be a number, uniformly distributed on the segment $ (0,\ 1) $, and let

$$ \omega \ = \ \sum _ {j=1} ^ \infty \frac{X _ {j} ( \omega )}{2 ^ j} , $$

where $ X _ {j} ( \omega ) = 0 $ or 1, be the expansion of $ \omega $ into a binary fraction. Then the $ X _ {j} $, $ j = 1,\ 2 \dots $ are independent and assume the values 0 and 1 with probability $ 1/2 $ each, i.e. the succession of zeros and ones in the expansion of $ \omega $ is described by the Bernoulli trial scheme with $ p = 1/2 $. However, the measure on $ (0,\ 1) $ can also be specified so as to obtain Bernoulli trials with any desired $ p $( if $ p \neq 1/2 $ the measure obtained is singular with respect to the Lebesgue measure).

Bernoulli trials are often treated geometrically (cf. 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

[G] B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian)
[F] W. Feller, "An introduction to probability theory and its applications", Wiley (1957–1971)
[K] M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1963) MR0110114 Zbl 0112.09101
How to Cite This Entry:
Bernoulli trials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_trials&oldid=44386
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article