Binomial Distribution · dandruce.co.uk

Parameters

n trials 20

p success 0.50

Mean np

10.0

Variance np(1-p)

5.0

Std dev

2.24

Mean, variance and standard deviation

For X ~ B(n, p):

$$E(X) = np \qquad \mathrm{Var}(X) = np(1-p)$$

The standard deviation is the square root of the variance, √(np(1-p)). As n grows the bars cluster more tightly around the mean (relative to n).

Full probability table

P(X = k) for each number of successes k

Parameters

n trials

p success

Probability of

Result

Working

Using a calculator

Most exam calculators give Binomial PD for P(X = k) and Binomial CD for P(X ≤ k). Build the rest from these:

You want	Use
P(X = k)	Binomial PD
P(X ≤ k)	Binomial CD
P(X < k)	CD at k−1
P(X ≥ k)	1 − CD at k−1
P(X > k)	1 − CD at k
P(a ≤ X ≤ b)	CD at b − CD at a−1

Casio ClassWiz

MENU → Distribution → Binomial PD or Binomial CD → choose Variable, then enter x (= k), N (= n) and p.

Highlighted bars show the probability you asked for

The binomial probability formula

Probability mass function

$$P(X = k) = \binom{n}{k}\, p^{k} (1-p)^{\,n-k}$$

where the binomial coefficient counts the number of ways to choose k successes from n trials:

$$\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$$

p is the probability of success on each trial and (1 - p) is the probability of failure. The term p^k(1-p)^n-k is the probability of one particular arrangement of k successes and n-k failures; the coefficient counts how many such arrangements there are.

When can I use the binomial distribution?

Fixed number of trials (n) - you decide n in advance.
Two outcomes - each trial is a success or a failure.
Independent trials - one trial does not affect any other.
Constant probability (p) - p is the same on every trial.

Mean and variance

$$E(X) = np \qquad \mathrm{Var}(X) = np(1-p)$$

These come from X being the sum of n independent Bernoulli trials, each with mean p and variance p(1-p).

Normal approximation

When n is large and p is not too close to 0 or 1, the binomial is well approximated by a normal distribution with the same mean and variance:

$$X \approx N\!\big(np,\ np(1-p)\big)$$

A common rule of thumb is that this works well when np > 5 and n(1-p) > 5. Use a continuity correction when moving between the discrete bars and the continuous curve.