Binomial Probability and Distributions
Big Idea: A binomial distribution models the number of successes in a fixed number of independent trials, where each trial has exactly two outcomes (success or failure) with the same probability of success.
1. What Makes an Experiment "Binomial"?
A binomial experiment must satisfy all four of these conditions:
① Fixed Number of Trials
The experiment consists of a fixed number $n$ of identical trials.
Example: Flipping a coin exactly 10 times.
② Two Outcomes Per Trial
Each trial results in one of two outcomes: Success (S) or Failure (F).
Example: Heads (success) or Tails (failure).
③ Constant Probability
The probability of success $p$ remains the same for every trial. The probability of failure is $q = 1 - p$.
Example: $P(\text{Heads}) = 0.5$ on every flip.
④ Independent Trials
The outcome of one trial does not affect the outcome of any other trial.
Example: Each coin flip is independent of the others.