View source for Bernoulli trials

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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.

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|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

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

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.).

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

$$ 
\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|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====
{|
|valign="top"|{{Ref|G}}|| B.V. Gnedenko,   [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint (1962) (Translated from Russian)
|-
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its   applications"|"An introduction to probability theory and its   applications"]], Wiley (1957–1971)
|-
|valign="top"|{{Ref|K}}|| M. Kac, "Statistical independence in probability, analysis and number theory" , Math. Assoc. Amer. (1963) {{MR|0110114}} {{ZBL|0112.09101}}
|}