In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.[1] It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his (1713).[2]
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:
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 (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:
Independent repeated trials of an experiment with exactly two possible outcomes are called Bernoulli trials. Call one of the outcomes "success" and the other outcome "failure". Let
p
q
p=1-q, q=1-p, p+q=1.
Alternatively, these can be stated in terms of odds: given probability
p
q
p:q
q:p.
of
oa
\begin{align} of&=p/q=p/(1-p)=(1-q)/q\\ oa&=q/p=(1-p)/p=q/(1-q). \end{align}
of=1/oa, oa=1/of, of ⋅ oa=1.
In the case that a Bernoulli trial is representing an event from finitely many equally likely outcomes, where
S
F
S:F
F:S.
\begin{align} p&=S/(S+F)\\ q&=F/(S+F)\\ of&=S/F\\ oa&=F/S. \end{align}
Random variables describing Bernoulli trials are often encoded using the convention that 1 = "success", 0 = "failure".
Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number
n
p
B(n,p)
k
B(n,p)
P(k)={n\choosek}pkqn-k
where
{n\choosek}
Bernoulli trials may also lead to negative binomial distributions (which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen), as well as various other distributions.
When multiple Bernoulli trials are performed, each with its own probability of success, these are sometimes referred to as Poisson trials.[3]
Consider the simple experiment where a fair coin is tossed four times. Find the probability that exactly two of the tosses result in heads.
For this experiment, let a heads be defined as a success and a tails as a failure. Because the coin is assumed to be fair, the probability of success is
p=\tfrac{1}{2}
q
q=1-p=1-\tfrac{1}{2}=\tfrac{1}{2}
Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by:
\begin{align} P(2) &={4\choose2}p2q4-2\\ &=6 x \left(\tfrac{1}{2}\right)2 x \left(\tfrac{1}{2}\right)2\\ &=\dfrac{3}{8}. \end{align}
What is probability that when three independent fair six-sided dice are rolled, exactly two yield sixes?
On one die, the probability of rolling a six,
p=\tfrac{1}{6}
q=1-p=\tfrac{5}{6}
As above, the probability of exactly two sixes out of three,
\begin{align} P(2) &={3\choose2}p2q3-2\\ &=3 x \left(\tfrac{1}{6}\right)2 x \left(\tfrac{5}{6}\right)1\\ &=\dfrac{5}{72} ≈ 0.069. \end{align}