**Exhibit 33.10 **Binomial distribution for n=28, p=0.4.

The binomial distribution models the number of successes in a sample of size
*n* drawn with replacement from a population of size *N*. If the sampling is carried out
without replacement, as is usually the case in research programmes, the draws are not independent.
However, where the population *N* is much larger (at least 10 times larger) than *n*, the
binomial distribution is a good approximation.

If the random variable *X* follows the binomial distribution with parameters
*n ∈ N* and *p ∈ [0,1]*, we write *X ~ B(n, p)*. The probability of getting exactly
*k* successes in *n* trials is given by the probability mass function (Exhibit 33.10):

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

Where:

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

The mean and variance of this distribution is:

$$mean,\, E(X)= \bar{p} = µ_X = np $$
$$variance,\, Var(X)=σ_X^2= np(1-p)$$

*Note: In Excel you may use the BINOM.DIST (k, n, p, cumulative) function to compute
P(k). *