Bernoulli trial

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In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure".

In practice it refers to a single experiment which can have one of two possible outcomes. These events can be phrased into "yes or no" questions:

  • Did the coin land heads?
  • Was the newborn child a girl?

Therefore success and failure are labels for outcomes, and should not be construed literally. 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.
  • Rolling a die, where a six is "success" and everything else a "failure".
  • In conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.

Contents

[edit] Definition

Independent repeated trials of an experiment with two outcomes only are called Bernoulli trials. Call one of the outcomes success and the other outcome failure. Let p be the probability of success in a Bernoulli trial. Then the probability of failure q is given by

q = 1 − p.

A binomial experiment consisting of a fixed number n of trials, each with a probability of success p, is denoted by B(n,p). The probability of exactly k successes in the experiment B(n,p) is given by:

P(k)={n \choose k} p^k q^{n-k}.

The function P(k) for k=0,1,\ldots,n for B(n,p) is called a binomial distribution.

Bernoulli trials may also lead to negative binomial, geometric, and other distributions as well.

[edit] Example: Tossing Coins

Consider the simple experiment where a fair coin is tossed four times. Find the probability that exactly two of the tosses result in heads.

[edit] Solution

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 = 1 / 2. Thus the probability of failure is given by

q = 1 − p = 1 − 1 / 2 = 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 \choose 2} p^2 q^2 \\
  &= 6 \times (1/2)^2 \times (1/2)^2 \\
  &= 3/8

\end{align}.

[edit] See also

[edit] References

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