## COUNTING

At the end of this section, students should be able to:

- state the principles of counting;
- find the number of ways of arranging \(n\) distinct objects;
- find the number of ways of arranging \(n\) objects some of which are identical;
- find the number of ways of choosing \(r\) distinct objects from a set of n distinct objects;
- identify a sample space;
- identify the numbers of possible outcomes in a given sample space;
- use Venn diagrams to illustrate the principles of counting;
- use possibility space diagram to identify a sample space;
- define and calculate the probability of event \(A, P(A)\), as the number of outcomes of \(A\) divided by the total number of possible outcomes, when all outcomes are equally likely and the sample space is finite;
- use the fact that \(0≤P(A)≤1\)
- demonstrate and use the property that the total probability for all possible outcomes in the sample space is 1;
- use the property that \(P(A')=1-P(A)\), where \(P(A')\) is the probability that event \(A\) does not occur
- use the property that \(P(A∪B)=P(A)+P(B)-P(A∩B)\), for event \(A\) and \(B\);
- use the property \(P(A∩B)=0\) or \(P(A∪B)=P(A)+P(B)\), where \(A\) and \(B\) are mutually exclusive events;
- use the property \(P(A∩B)=P(A)×P(B)\) or \(P(A|B)=P(A)\) where \(A\) and \(B\) are independent events;
- use the property \(\displaystyle P(A|B)={P(A∩B)\over P(B)}\) where \(P(B)≠0\).
- use a tree diagram to list all possible outcomes for conditional probability.