Kategorier

## Combinations

It is called acombination of r elements if the order of the elements in a selection of r elements out of n elements is irrelevant and redundance is not allowed. Another way of ststing this is to say that all elements are selected at once and not one-by-one.

This number is given by the ratio n!/(n-r)!. It is also necessary to divide by r!  since redundance is not allowed.

Ex. The capital of Madagascar is called ANTANANARIVE.

The number of letters here is 12 but to find all permutations we need to take into consideration that we have three N therefore we get the same word independently of their internal order and must theefore divide by 3!. The same is the case for the four A which force us to divide by 4!.

The number of combinations for this word therefore is 12!/(4!3!).

Generally the number of subsets with r components selected out of n elements is

C(n,r) = n!/((n-r)!r!)

Kategorier If you are in a situation where you have two make two consecutive choices and the first one can be selected from n alternatives and the second can be selected from m alternatives the total number of possible combinations is n*m. This can be easilty understood since for each of the n choices of A there are m possibilities to select the second item.

Ex: Då man väljer en Golf cabriolet kan man välja mellan fyra lackfärger, fyra klädselfärger och fyra fyra motorer.
Hur många olika varianter av Golf cabriolet är teoretiskt möjliga?

Svar: 4 * 4 * 4 = 64 st.enligt multilikationsprincipen.

Ex. Determine the number of sub-sets to a set consisting of 10 elements.

A subset of a set M can be determined by going through the elementsof M, one at a time, and determines whether it belongs to the subset or not. For each element there are then 2 options: either it belongs to the subset or it doesn’t. Hence the totoal number of possibilities is 210.

Therefore the total number of subsets for the set M is 210.

In the general case when M has n elements the number of subsets is 2n.

On the other hand if you are are going to choose one item from either subset A or from subset the number of possibilities you can choose from is obviously m + n.

Kategorier

## Venn-diagrams

A good way of illustrating probabilities is to use so-called Venn-diagrams. In effect this means representing the probability of an event with circles. Mutually excluding events can be represented by two separate non-overlapping ciecles.

P(A) + P(B) = P(A U B)

Simultanously events can be illustrated by partially overlapping circles. The overlap then represents the instance of both events happening simultanously.

Kategorier

## Probability

The probability of a certan outcome of an experiment can be calculated as the quotient of the number of outcomes giving the desired result and the number of possible outcomes.

P(A) = (number of outcomes giving the desired results)/ (number of all  possible outcomes)
Therfore 0 < P(A) < 1 with 0 being the probability for an impossible outcome and 1 the probability for a certain outcome.
A multi-step experiment can be illustrated with a tree-diagram.
Two outcomes are complementary if either of them most occur if the other doesn’t.
The sum of their probabilities must be one.
http://en.wikipedia.org/wiki/Probability_theory

Event Probability
A                P(A)
not A        CP(A) = 1- P(A) The complement to A
A or B      P(A U B)= P(A) + P(B) for mutually exclusive events otherwise

P(A U B)= P(A) + P(B) – P(A ∩ B) The union of A and B i.e. the occurence of either of them.

A and B   P(A ∩ B) The occutence of both of them at the same time can be found by multplying the individual probabilities: P(A)*P(B)*P(C)*…….. *P(Z).
A given B P(A¦B) =P(A ∩ B)/P(B)  this gives the probability for A given that B has already happened.