# Bernoulli trials

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