Binomial Probability Distributions

Binomial Probability Distributions: Conditions

• A procedure generates a binomial probability distribution if four conditions are met:

  1. Two Outcomes: Each trial has exactly two outcomes: Success and Failure. Define Success beforehand.

  2. **Fixed Number of Trials $(n)$: **The number of trials must be known and fixed.

  3. Independent Trials: Trials must be independent of each other (e.g., with replacement).

  4. **Constant Probability of Success $(p)$: **The probability of success must remain the same for each trial.

Components of a Binomial Procedure

• Success: The specific outcome of interest (an action, not a quantity).

• $n$: The total number of trials.

• $x$: The specific number of successes desired (must be a whole number).

• $p$: The probability of success in a single trial $(0 \leq p \leq 1)$.

• $q$: The probability of failure in a single trial, calculated as $q = 1 - p$ .