The binomial theorem - the missing link
If you have gone to the links on permutations and combinations you will know how to work out the number of different ways there are to choose r items from a total of n items (combinations) and how many arrangements they make (permutations). But these techniques do not tell us how to start deciding howlikely the various choices are. If you went to the link on probability then you will know how to start working out the probability of an event occurring.
The binomial theorem links
to give the likelihood of a series of events.
Using combinations to get to the probability of certain events occurring: the binomial theorem
The
binomial theorem is used in several areas of mathematics, including probability.
To explain how it works we are going to continue to use and expand the example
given in the probability document. Consider the example given: choosing
cards over two rounds, with a probability of one-fifth (
)
of getting the correct card in each round.
The binomial theorem enables one to obtain the probability of an event or a number of events. Below is a general binomial equation for an event:
In the equation n is the number of rounds, r is the number of rounds in which a matching card was chosen, p is the probability of choosing a matching (correct) card and q is the probability of choosing an incorrect card.
Looking back at combinations and probability we can see that the fraction part of the general equation is the number of combinations of choosing a particular path:
and the remainder of the general equation is the probability of getting the particular score along ONE of these paths: prq(n – r).
When we multiply the probability of getting our score from one path by the number of possible paths, (i.e. adding the probability of all the branches together) we get the probability of getting our score.
probability = number of branches × probability of getting the score along one branch of a probability tree