Bernoulli trial
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 Ars Conjectandi (1713).
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.
Examples of Bernoulli trials include: Suppose there exists an experiment consiting of indepently repeated trials, each of which has only two possible outcomes; called experimental Bernoulli trials.
experimental realizations of success (1) and failure (0) will be defined by a Bernoulli random variable:
be the probability of success in a Bernoulli trial, and
Then the probability of success and the probability of failure sum to one, since these are complementary events: "success" and "failure" are mutually exclusive and exhaustive.
Thus, one has the following relations: Alternatively, these can be stated in terms of odds: given probability
These can also be expressed as numbers, by dividing, yielding the odds for,
: These are multiplicative inverses, so they multiply to 1, with the following relations: In the case that a Bernoulli trial is representing an event from finitely many equally likely outcomes, where
This yields the following formulas for probability and odds: Here the odds are computed by dividing the number of outcomes, not the probabilities, but the proportion is the same, since these ratios only differ by multiplying both terms by the same constant factor.
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
of statistically independent Bernoulli trials, each with a probability of success
A random variable corresponding to a binomial experiment is denoted by
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.
[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
, is given by Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by: 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,
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 .
Grey curve : To get 50-50 chance of throwing a Yahtzee (5 cubic dice all showing the same number) requires 0.69 × 1296 ~ 898 throws.
Green curve : Drawing a card from a deck of playing cards without jokers 100 (1.92 × 52) times with replacement gives 85.7% chance of drawing the ace of spades at least once.
