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Statistics · Statistical distributions

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The binomial distribution as a model

When the count of "successes" in a fixed number of repeated trials follows a binomial distribution X ~ B(n, p): the four conditions (a fixed number of trials, two outcomes per trial, a constant probability of success, and independent trials), how to identify n and p from a context, how to state a condition for a binomial model, and when the binomial is NOT appropriate.

Statistics · Statistical distributions

The binomial distribution as a model

When the count of "successes" in a fixed number of repeated trials follows a binomial distribution X ~ B(n, p): the four conditions (a fixed number of trials, two outcomes per trial, a constant probability of success, and independent trials), how to identify n and p from a context, how to state a condition for a binomial model, and when the binomial is NOT appropriate.

Why it works

Many situations are the same experiment repeated over and over, where each repeat either "succeeds" or "fails". If you count the number of successes, that count often follows a binomial distribution, written

X∼B(n,p),X \sim B(n, p),

where nn is the number of trials and pp is the probability of success on each trial. (So in B(15,0.48)B(15, 0.48), there are n=15n = 15 trials each with success probability p=0.48p = 0.48.)

The four conditions for a binomial model. XX can be modelled by B(n,p)B(n, p) when:
  1. there is a fixed number of trials, nn;
  2. each trial has only two outcomes — "success" or "failure";
  3. the probability of success, pp, is constant for every trial;
  4. the trials are independent of one another.
If a question asks you to "state a condition for XX to be modelled by B(n,p)B(n,p)", you quote one of these — most often that the trials are independent or that pp is constant.

Identifying nn and pp. Read the context: nn is how many times the trial is repeated and pp is the probability of the outcome you are counting. "8%8\% of students are members, a sample of 3636 is taken, XX is the number who are members" gives X∼B(36,0.08)X \sim B(36, 0.08).

Here is the shape of B(10,0.3)B(10, 0.3) — the height of each bar is P(X=r)P(X = r):2468100.050.10.150.20.25rP(X = r)The distribution is bunched near np=3np = 3 and tails off — when p<0.5p < 0.5 it is skewed to the right, when p=0.5p = 0.5 it is symmetric, and when p>0.5p > 0.5 it is skewed to the left.

When the binomial does not apply. Watch for a broken condition:
  • Sampling without replacement from a small group changes pp from trial to trial,
so pp is not constant (and trials are not independent) — not binomial.
  • No fixed nn (e.g. "keep rolling until a six") — not binomial.
  • More than two outcomes that you can't reduce to success/failure — not binomial.
(When sampling from a very large population, pp stays effectively constant, so a binomial model is still reasonable — which is why "a random sample of 3636 from a whole university" can be modelled binomially.)