What Is the Binomial Distribution?
The binomial distribution is one of the most important discrete probability distributions in statistics. It models situations where you repeat an experiment a fixed number of times and count how many times a particular outcome (called a "success") occurs.
Binomial Distribution: A probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes and the probability of success remains constant throughout.
The name binomial comes from the two outcomes ("bi" = two) involved in each trial , success or failure. This links directly to the binomial theorem you may have encountered in algebra.
Think of flipping a biased coin 20 times and counting how many heads you get. Each flip is independent, there are only two outcomes (heads or tails), the probability stays the same every time, and there are a fixed number of flips. That's a perfect binomial setup.
The Four Conditions for a Binomial Experiment
Before applying the binomial distribution, you must verify that all four of the following conditions are satisfied:
- Fixed number of trials , the experiment is repeated a set number of times, denoted .
- Two outcomes only , each trial results in either a "success" or a "failure" (the labels are arbitrary).
- Constant probability , the probability of success, denoted , is the same for every trial.
- Independent trials , the outcome of one trial does not affect the outcome of any other.
If any of these conditions fails, the binomial distribution is not the correct model.
A very common mistake is applying the binomial distribution to sampling without replacement from a small population. When items are removed from the pool, the probability of success changes with each draw , violating condition 3. For example, drawing cards from a standard deck without replacing them is not binomial.
Is this binomial?
Scenario A: Rolling a die 15 times and counting how many times you roll a 6.
- Fixed trials: ✓ ()
- Two outcomes: ✓ (roll a 6 = success; anything else = failure)
- Constant probability: ✓ ( each roll)
- Independent: ✓ (dice rolls don't affect each other)
→ Yes, binomial.
Scenario B: Drawing 5 cards from a deck (without replacement) and counting aces.
- Constant probability: ✗ (probability changes after each draw)
→ Not binomial.