Bernoulli trials

2020 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