Probability Distribution of Bernoulli Trials

T. Dhasaratharaman*

Statistician, Kauvery Hospitals, India

*Correspondence: Tel.: +91 90037 84310; email:

A Bernoulli experiment is a random experiment, the outcome of which can be classified in one of two mutually exclusive and exhaustive ways, mainly, Success or Failure (e.g., female or male, life or death, non-defective or defective).

So, it is also called a Binomial trial!

A sequence of Bernoulli trials occurs when a Bernoulli experiment is performed several independent times so that the probability of success, say, p, remains the same from trial to trial. That is, in such a sequence we let p denote the probability of success on each trial. In addition, frequently q=1-p denote the probability of failure; that is, we shall use q and 1-p interchangeably.

Let X be a random variable associated with Bernoulli trial by defining it as follows:

X (success) = 1 and X (failure) = 0. That is the two outcomes, success and failure, is denoted by one and zero, respectively. It can be written as f(x) =px(1-p) 1-x and we say that X has a Bernoulli distribution.


  1. A Bernoulli (Success-Failure) experiment is performed n times.
  2. The trials are independent.
  3. The probability of success on each trial is a constant p; the probability of failure is q=1-p.
  4. The random variable X counts the number of successes in the n trials.

A binomial distribution will be denoted by the symbol b(n,p) and we say that the distribution of X is b(n,p). The constants n and p are called the parameters of the binomial distribution, they correspond to the number n of independent trials and the probability p of success on each trial.

A random variable X will have Bernoulli distribution with probability p if its probability distribution is

P(X = x) = px (1 – p)1-x, for x = 0, 1 and P(X = x) = 0 for other values of x.

Here, 0 is failure and 1 is the success.

Example 1: In a hospital, a particular type of operation has 95% chance of success. On a given day, 10 operations were performed. This can be viewed as 10 Bernoulli trials, with each trial having a probability of success p=0.95, and a probability of failure 1-p=0.05.


Mr. T. Dhasaratharaman