Statisticsdrill

Distributions Updated November 2026

Binomial distribution

The binomial distribution gives the probability of exactly k successes in n independent trials that each succeed with probability p: P(X = k) = nCk * p^k * (1 - p)^(n - k). Its mean is np and its standard deviation is sqrt(np(1 - p)).

Syntax

P(X = k) = nCk * p^k * (1 - p)^(n - k)
PartWhat it is
nthe number of trials
pthe probability of success on each trial
kthe number of successes you want the probability of
nCkthe number of ways to place k successes among n trials

Examples

10 coin flips, exactly 3 heads
P = 10C3 * 0.5^3 * 0.5^7 = 120 / 1024

Result: 0.1172

Excel: =BINOM.DIST(3, 10, 0.5, FALSE)

3 free throws at p = 0.2, at least one made
1 - P(0) = 1 - 0.8^3 = 1 - 0.512

Result: 0.488

Excel: =1 - BINOM.DIST(0, 3, 0.2, FALSE)

n = 100, p = 0.2
mean = 100 * 0.2; sd = sqrt(100 * 0.2 * 0.8) = sqrt(16)

Result: mean = 20, sd = 4

At most 2 successes in 5 trials, p = 0.3
=BINOM.DIST(2, 5, 0.3, TRUE)

Result: 0.8369

Common mistakes

  • Using the binomial when trials are not independent or p changes, such as drawing without replacement; use the hypergeometric distribution there.
  • Mixing up FALSE and TRUE in BINOM.DIST; FALSE gives exactly k, TRUE gives k or fewer.
  • Computing at least k as 1 - P(k); at least k means 1 - P(X <= k - 1).

Practice Binomial distribution

These drills check your answer on live data, free, no account needed.

  • Binomial: exactly kHistorically 10% of sales calls end in a sale. Out of the next 8, independent of each other, what is the probability that exactly 3 end in a sale? Round to 4 decimal places.
  • Binomial: at least kA marketing email gets a click from 15% of recipients. It goes to 10 recipients who act independently. What is the probability that at least 5 of them click? Round to 4 decimal places.
  • Binomial standard deviationA call center closes a sale on 8% of calls. Over 40 independent calls, what is the standard deviation of the number of sales? Round to 2 decimal places.

Questions

What are the conditions for a binomial distribution?

A fixed number of trials, each trial independent, only two outcomes per trial, and the same success probability on every trial.

When can I approximate the binomial with the normal?

A common rule is np >= 10 and n(1 - p) >= 10 (some texts use 5). Then X is roughly normal with mean np and sd sqrt(np(1 - p)); add a 0.5 continuity correction for better accuracy.

Related

  • Permutations and combinationsWhen order matters use permutations nPr = n! / (n - r)!; otherwise combinations nCr = n! / (r! (n - r)!). Examples plus Excel PERMUT and COMBIN.
  • Poisson distributionThe Poisson formula P(X = k) = e^(-lambda) lambda^k / k! for counts of events per interval, worked examples with exact results and Excel POISSON.DIST.
  • Normal distributionHow to find normal distribution probabilities with z-scores, read the standard normal table, find a percentile cutoff, and use NORM.DIST and NORM.INV.
  • Expected valueExpected value is the probability-weighted average of all outcomes. Formula, worked examples with dice, raffles and bets, and the Excel SUMPRODUCT formula.

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