Bernoulli Trials

Suppose we have a jar with \(7\) marbles, four of which are red and three are blue. A marble is drawn at random and we record whether it is red or blue. The probability \(p\) of getting a red marble is \(4/7\); and the probability of getting a blue is \(1-p=3/7\).

Now suppose the marble is put back in the jar, the marbles in the jar are stirred, and the experiment is repeated. Then the probability of getting a red marble on the second trial is again \(4/7\), and this pattern holds regardless of the number of times the experiment is repeated.

It is customary to call this situation a series of Bernoulli trials. More formally, we have an experiment with only two outcomes: success and failure. The probability of success is \(p\) and the probability of failure is \(1-p\). Most importantly, when the experiment is repeated, then the probability of success on any individual test is exactly \(p\).

We fix a positive integer \(n\) and consider the case that the experiment is repeated \(n\) times. The outcomes are then the binary strings of length \(n\) from the two-letter alphabet \(\{S,F\}\), for success and failure, respectively. If \(x\) is a string with \(i\) successes and \(n-i\) failures, then \(P(x)=\binom{n}{i}p ^i(1-p)^{n-i}\). Of course, in applications, success and failure may be replaced by: head/tails, up/down, good/bad, forwards/backwards, red/blue,

Condensed: the full section is in Keller & Trotter, Applied Combinatorics.