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

In this case the

possible outcomes are easily enumerated: (rrbb), (rbrb), (brrb), (rbbr), (brbr), (bbrr).

(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 nth draw or draw i − 1 red balls in the first n − 1 draws followed by the ith red ball on the nth draw. Hence,

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