Bernoulli Trials and Probability Models

Bernoulli Trials

Bernoulli trials are a foundational concept in probability and statistics, describing experiments with exactly two possible outcomes. These outcomes are often labeled as "success" and "failure." Bernoulli trials are the basis for several important probability models.

• Definition: A collection of real or simulated data are called Bernoulli trials if:

  • Each trial (observation) has exactly two possible outcomes (success or failure).

  • The probability of success is the same for each trial.

  • The trials are independent.

10% Condition

• When sampling without replacement, trials are not strictly independent. However, it is acceptable to treat them as independent if the sample size is less than 10% of the population.

Geometric Probability Model

The geometric probability model is used for random variables that count the number of Bernoulli trials until the first success occurs.

• Probability of success on the nth trial:

• Mean (expected value):

• Standard deviation:

Binomial Probability Model

The binomial probability model is appropriate for random variables that count the number of successes in a fixed number of Bernoulli trials.

• Probability of exactly k successes in n trials:

• Mean (expected value):

• Standard deviation:

Success/Failure Condition

• For a normal model to be a good approximation of a binomial model, both and must be at least 10. That is, and .

Statistical Significance

• A result is considered statistically significant if it has a very low probability of occurring by chance alone.

Example Applications

• Bernoulli Trials: Flipping a coin (success = heads, failure = tails).

• Geometric Model: Counting the number of coin flips until the first heads appears.

• Binomial Model: Counting the number of heads in 10 coin flips.