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{{MSC|60F05}} 

[[Category:Limit theorems]]

The (historically) original form of the (weak) [[Law of large numbers|law of large numbers]]. The theorem appeared in the fourth part of Jacob Bernoulli's book Ars conjectandi (The art of conjecturing). This part may be considered as the first serious study ever of probability theory. The book was published in 1713 by N. Bernoulli (a nephew of Jacob Bernoulli). The theorem deals with sequences of independent trials, in each one of which the probability of occurrence of some event ( "success" ) is  $  p $. 
Let  $  n $
be the number of trials and let  $  m $
be the random variable equal to the number of successful events. The Bernoulli theorem states that, whatever the value of the positive numbers  $  \epsilon $
and  $  \eta $, 
the probability  $  {\mathsf P} $
of the inequality

$$ 
- \epsilon  \leq   
\frac{m}{n}
 - p  \leq   \epsilon
$$

will be higher than  $  1 - \eta $
for all sufficiently large  $  n $(
$  n \geq  n _ {0} $). 
The proof of this theorem, which was given by Bernoulli and which was exclusively based on a study of the decrease of probabilities in the binomial distribution as one moves away from the most probable value, was accompanied by an inequality which made it possible to point out a certain bound for the given  $  n _ {0} $
if  $  \epsilon $
and  $  \eta $
were given. Thus, it was found by Bernoulli that if  $  p = 2/5 $, 
the probability of the inequality

$$ 
{- 
\frac{1}{50}
 }  \leq   {
\frac{m}{n}
 } -
{
\frac{2}{5}
 }  \leq   {
\frac{1}{50}
 }
$$

will be higher than 0.999 if  $  n \geq  25,550 $. 
By introducing a slight improvement in the original reasoning of Bernoulli, it is possible to conclude that it is sufficient to select a value of  $  n $
obeying the condition

$$ 
n  >  
\frac{1 + \epsilon }{\epsilon  ^ {2} }
 \ 
{ \mathop{\rm log}  
\frac{1} \eta 
 + 
\frac{1} \epsilon 
 } ,
$$

which gives in turn, for the probability  $  1 - {\mathsf P} $
of the inequality

$$ 
{\left | 
\frac{m}{n}
 - p \right | }  >  \epsilon ,
$$

an estimate of the form

$$ 
2   \mathop{\rm exp} \left \{ - {
\frac{1}{2}
 } n \epsilon  ^ {2} \right \} .
$$

The condition obtained for the above example is  $  n \geq  17,665 $(
more sophisticated estimates show that it is sufficient to take  $  n \geq  6502 $; 
one may note, for the sake of comparison, that the de Moivre–Laplace theorem yields 6498 as the approximate value of  $  n _ {0} $). 
Other estimates for  $  1 - {\mathsf P} $
may be obtained using the [[Bernstein inequality|Bernstein inequality]] and its analogues. See also [[Binomial distribution|Binomial distribution]].

====References====
{|
|valign="top"|{{Ref|Bi}}|| J. Bernoulli, "Ars conjectandi" , ''Werke'' , '''3''' , Birkhäuser (1975) pp. 107–286 (Original: Basle, 1713) {{MR|2195221}} {{ZBL|0365.01016}}
|-
|valign="top"|{{Ref|M}}|| A.A. Markov, "Wahrscheinlichkeitsrechung" , Teubner (1912) (Translated from Russian) {{MR|}} {{ZBL|}}
|-
|valign="top"|{{Ref|Bn}}|| S.N. Bernshtein, "Probability theory" , Moscow-Leningrad (1946) (In Russian) {{MR|}} {{ZBL|53.0492.01}}
|}

====Comments====

====References====
{|
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1''' , Wiley (1965)
|-
|valign="top"|{{Ref|S}}|| R.J. Serfling, "Approximation theorems of mathematical statistics" , Wiley (1980) pp. 6, 96 {{MR|0595165}} {{ZBL|0538.62002}}
|}