Editing Bernoulli trial (section)

{{short description|Any experiment with two possible random outcomes}}
{{Probability fundamentals}}
[[File:Bernoulli_trial_progression.svg|thumb|400px|Graphs of probability ''P'' of not observing independent events each of probability ''p'' after ''n'' Bernoulli trials vs ''np'' for various ''p''. Three examples are shown:
<br />'''Blue curve''': Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as ''n'' increases, the probability of a 1/''n''-chance event never appearing after ''n'' tries rapidly converges to ''1/e''.
<br />'''Grey curve''': To get 50-50 chance of throwing a [[Yahtzee]] (5 cubic dice all showing the same number) requires 0.69 &times; 1296 ~ 898 throws.
<br />'''Green curve''': Drawing a card from a deck of playing cards without jokers 100 (1.92 &times; 52) times with replacement gives 85.7% chance of drawing the ace of spades at least once.]]
In the theory of [[probability]] and [[statistics]], a '''Bernoulli trial''' (or '''binomial trial''') is a random [[Experiment (probability theory)|experiment]] with exactly two possible [[Outcome (probability)|outcomes]], "success" and "failure", in which the probability of success is the same every time the experiment is conducted.<ref>{{cite encyclopedia | last = Papoulis | first =  A. | contribution = Bernoulli Trials | title = Probability, Random Variables, and Stochastic Processes | edition =  2nd | location = New York | publisher = [[McGraw-Hill]] | pages = 57–63 | year = 1984}}</ref> It is named after [[Jacob Bernoulli]], a 17th-century Swiss mathematician, who analyzed them in his ''{{Lang|la|[[Ars Conjectandi]]}}'' (1713).<ref>James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45</ref>

The mathematical formalization and advanced formulation of the Bernoulli trial is known as the [[Bernoulli process]].

Since a Bernoulli trial has only two possible outcomes, it can be framed as a "yes or no" question. For example:

*Is the top card of a shuffled deck an ace?
*Was the newborn child a girl? (See [[human sex ratio]].)

Success and failure are in this context labels for the two outcomes, and should not be construed literally or as value judgments. More generally, given any [[probability space]], for any [[Event (probability theory)|event]] (set of outcomes), one can define a Bernoulli trial according to whether the event occurred or not (event or [[complementary event]]). Examples of Bernoulli trials include:

*[[Flipping a coin]]. In this context, obverse ("heads") conventionally denotes success and reverse ("tails") denotes failure. A [[fair coin]] has the probability of success 0.5 by definition. In this case, there are exactly two possible outcomes.
*Rolling a die, where a six is "success" and everything else a "failure". In this case, there are six possible outcomes, and the event is a six; the complementary event "not a six" corresponds to the other five possible outcomes.
*In conducting a political [[opinion poll]], choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.