# Applications of simple probability experiments

## Multinomial probability

A basic problem first solved by Jakob Bernoulli is to find the probability of obtaining exactly *i* red balls in the experiment of drawing *n* times at random with replacement from an urn containing *b* black and *r* red balls. To draw at random means that, on a single draw, each of the *r* + *b* balls is equally likely to be drawn and, since each ball is replaced before the next draw, there are (*r* + *b*) ×⋯× (*r* + *b*) = (*r* + *b*)^{n} possible outcomes to the experiment. Of these possible outcomes, the number that is favourable to obtaining *i* red balls and *n* − *i* black balls in any one particular order is

The number of possible orders in which *i* red balls and *n* − *i* black balls can be drawn from the urn is the binomial coefficient

where *k*! = *k* × (*k* − 1) ×⋯× 2 × 1 for positive integers *k*, and 0! = 1. Hence, the probability in question, which equals the number of favourable outcomes divided by the number of possible outcomes, is given by the binomial distribution

where *p* = *r*/(*r* + *b*) and *q* = *b*/(*r* + *b*) = 1 − *p*.

For example, suppose *r* = 2*b* and *n* = 4. According to equation (3), the probability of “exactly two red balls” is

In this case the

possible outcomes are easily enumerated: (*r**r**b**b*), (*r**b**r**b*), (*b**r**r**b*), (*r**b**b**r*), (*b**r**b**r*), (*b**b**r**r*).

(For a derivation of equation (2), observe that in order to draw exactly *i* red balls in *n* draws one must either draw *i* red balls in the first *n* − 1 draws and a black ball on the *n*th draw or draw *i* − 1 red balls in the first *n* − 1 draws followed by the *i*th red ball on the *n*th draw. Hence,

from which equation (2) can be verified by induction on *n*.)