Bernoulli Distribution


< Probability Distributions List

A Bernoulli distribution is a discrete probability distribution for a Bernoulli trial — a random experiment that has only two possible outcomes (usually called a “success” or a “failure”). The probability of success is P and the probability of failure is 1 – P. For example, the probability of getting a heads (a success) while flipping a coin is 0.5. The probability of failure is 0.5 as well (1 – 0.5 = 0.5).

In this example, the probability of success (1) is 0.4, and the probability of failure (0) is 0.6:

A Bernoulli distribution is a special case of the binomial distribution. It is a binomial distribution with a single trial (e.g. a single coin toss).

Watch Prof. Essa’s video for the definition of a Bernoulli distribution, the mean and variance.

The Mean and Variance of the Bernoulli Distribution


The mean of the Bernoulli distribution is equal to P, the probability of success. This makes sense intuitively — if you think about all the possible outcomes of a Bernoulli trial (e.g. all the possible outcomes of flipping a coin), the mean would be equal to the sum of all those probabilities, which is just P (the probability of success).

The variance of the Bernoulli distribution is also equal to P(1-P), or simply P-P^2 . This result can be derived by first calculating the variance of the binomial distribution and then substituting n=1 into that formula.

Bernoulli Trial

A Bernoulli trial is one of the simplest experiments you can conduct. It’s an experiment where there are two possible outcomes, like “Yes” and “No.” A few examples:

  • Coin Tossing – record how many coins land heads up or tails down? Births- what percent boys were born on any given day compared to girls (or vice versa)?
  • Rolling Dice – does having more successes mean better luck for certain rolls as opposed to others?

One of the most important aspects about Bernoulli trials is that each action must be independent. This means you cannot depend on what happened before, because it will affect your future outcomes–for example if I winning a scratch off lottery ticket then my odds would change from any other similar tickets being drawn as well. Dependent events such as drawing lotto numbers come with different probabilities depending upon how many balls remain in play; when there are 100 left they have 1/100th chance but when there are only ten balls left, the probability increases to 1/10.