Binomial Distribution

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.

The random variable

Let

Then

and possible values are $0,1,2,\ldots,n$.

The word “success” means the outcome being counted; it need not be desirable.

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.

Cumulative probabilities

Translate language carefully:

• exactly $k$: $P(X=k)$;
• at most $k$: $P(X\le k)$;
• fewer than $k$: $P(X<k)=P(X\le k-1)$;
• at least $k$: $P(X\ge k)$;
• more than $k$: $P(X>k)=P(X\ge k+1)$.

Use a calculator distribution function or add the appropriate exact probabilities.

Complement method

“At least one” is usually simplest through

More generally,

Check whether equality belongs on the subtracted side.

Mean and variance

For $X\sim\operatorname{Bin}(n,p)$,

The mean is an expected count and need not be an integer.

Shape

When $p=0.5$, the distribution is symmetric. Small $p$ produces right skew, and large $p$ produces left skew. As $n$ grows and both $np$ and $n(1-p)$ become sufficiently large, a normal approximation can become useful.

The exact binomial model remains discrete.

Independence and sampling

Sampling without replacement is not exactly independent because each draw changes the population. If the sample is small compared with a large population, independence may be a reasonable approximation; otherwise a hypergeometric model is more appropriate.

State assumptions rather than automatically using binomial formulas.

Technology

Calculator functions often distinguish binompdf for exactly one value and binomcdf for cumulative probability from $0$ through a value. Translate the event before selecting a command.

Technology evaluates the model; it does not verify the BINS conditions.

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?

Related topics

• Combinations
• Random Variables
• Expected Value
• Normal Distribution