Bernoulli Trials
A powerful framework for understanding experiments with exactly two outcomes.
📌 Formal Definition
A Bernoulli Trial is a random experiment that has exactly two possible outcomes:
- Success (S) — with probability $ p $
- Failure (F) — with probability $ q = 1 - p $
When we repeat a Bernoulli trial $ n $ times independently (meaning the outcome of one trial doesn't affect another), we call this a sequence of Bernoulli Trials.
Key Properties:
- Each trial has exactly two outcomes (Success or Failure).
- The probability of success $ p $ remains constant for every trial.
- All trials are independent of each other.
- The number of trials $ n $ is fixed in advance.
🎯 Real-World Examples of Bernoulli Trials
| Experiment | Success (S) | Failure (F) | $ p $ |
|---|---|---|---|
| Flipping a fair coin | Heads | Tails | $ \frac{1}{2} $ |
| Rolling a die (want a 6) | Getting 6 | Not getting 6 | $ \frac{1}{6} $ |
| Free throw in basketball | Ball goes in | Ball misses | $ 0.75 $ (varies) |
| True/False quiz (guessing) | Correct answer | Wrong answer | $ \frac{1}{2} $ |