Math101 · Independent Ontario learning library
Math101Binomial Distribution
The binomial distribution models the number of successes in a fixed set of independent, equal-probability trials.
A binomial model counts successes—not their order—across repeated trials with stable conditions.
The BINS conditions
A binomial setting requires:
- Binary outcome: each trial is classified success/failure;
- Independent trials;
- fixed Number of trials $n$;
- Same success probability $p$ on every trial.
If any condition fails substantially, a different model may be needed.
Probability formula
The probability of exactly $x$ successes is
The combination chooses which $x$ trial positions are successes. The powers give the probability of one such success/failure pattern.
Worked example: exactly x successes
The factor ${10\choose4}$ accounts for every order containing four successes.
Complement method
“At least one” is usually simplest through
More generally,
Check whether equality belongs on the subtracted side.
Common mistakes
Using the model when $p$ changes between trials. Stable probability is required.
Omitting the combination coefficient. Many success orders lead to the same count.
Confusing $p$ and $1-p$. Match exponents to successes and failures.
Translating “at least” as $\le$. Write the event symbolically first.
Assuming sampling without replacement is independent. Check the population context.
Quick self-check
- Do binary, independent, fixed-$n$, same-$p$ conditions hold?
- What exactly does $X$ count?
- Are $n$, $p$, and $x$ identified correctly?
- Does the event ask for exact, cumulative, or complement probability?
- Are the combination and both probability powers included?
- Is the final probability between $0$ and $1$ and interpreted in context?
