Bernoulli trial

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.

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.

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.

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:

$\begin{align} P(2) &= {4 \choose 2} p^2 q^2 \\ &= 6 \times (1/2)^2 \times (1/2)^2 \\ &= 3/8 \end{align}$.

See also

• Bernoulli scheme
• Bernoulli sampling
• Bernoulli distribution
• Binomial distribution
• Binomial proportion confidence interval
• Poisson sampling
• Sampling design
• Coin flipping
• Jacob Bernoulli

References

• Weisstein, Eric W., "Bernoulli Trial" from MathWorld.