Inverse Bernoulli Sampling

TL;DR

• Choose the probability p of the desired (binary) outcome.
• Generate a random number u between 0 and 1 and compare it to p.
• If u < p, return the desired outcome; otherwise return the other outcome.

Definition

Inverse Bernoulli sampling is a method for randomly selecting a binary outcome with a specified probability.

Explanation

To perform inverse Bernoulli sampling, first specify the probability p of the desired outcome. Then generate a random number u between 0 and 1. Compare u to p: a value less than p corresponds to the desired outcome, and a value greater than or equal to p corresponds to the other outcome. This procedure simulates a binary random event according to the specified probability.

In inequality form:

$$u < p$$

indicates the desired outcome, and

$$u \ge p$$

indicates the alternative outcome.

Examples

Coin toss

• Set the probability of heads to 0.5.
• Generate a random number between 0 and 1 using the inverse Bernoulli distribution.
• If the random number is less than 0.5, output “heads”. Otherwise, output “tails”.

Medical trial

• Suppose the probability of the drug being effective is 0.75.
• Use inverse Bernoulli sampling to simulate outcomes for individual patients: a value less than 0.75 indicates the drug was effective for that patient, and a value greater than or equal to 0.75 indicates it was not.

Use cases

• Coin tosses
• Medical trials
• Other scenarios where a random binary event needs to be simulated

Related terms

• Inverse Bernoulli distribution
• Bernoulli distribution
• Binary outcome