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The simplest probability distribution is the Bernoulli distribution and it is associated with a single experiment with only two possible outcomes. An example of this type of experiment is the toss of a coin. Let X be a random variable representing the experiment, then X is a random variable that takes only two outcomes, 0 and 1, where X = 1 means that a favorable event is observed, and X = 0 otherwise. We assume that the probability of the favorable event, generally called the probability of success, is a real number p such that 0 ≤ p ≤ 1. The probability mass function p_X(x) is then:

(1.29)

The mean of the Bernoulli distribution is then μ_X = p and the variance is .

The Bernoulli distribution has several applications in earth sciences. In reservoir modeling, for example, we can use the Bernoulli distribution for the occurrence of a given facies or rock type. For instance, we define a successful event as finding a high‐porosity sand rather than impermeable shale. The probability of success is generally unknown and it depends on the overall proportions of the two facies.