## COUNTING

At the end of this section, students should be able to:
1. state the principles of counting;
2. find the number of ways of arranging $$n$$ distinct objects;
3. find the number of ways of arranging $$n$$ objects some of which are identical;
4. find the number of ways of choosing $$r$$ distinct objects from a set of n distinct objects;
5. identify a sample space;
6. identify the numbers of possible outcomes in a given sample space;
7. use Venn diagrams to illustrate the principles of counting;
8. use possibility space diagram to identify a sample space;
9. 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;
10. use the fact that $$0≤P(A)≤1$$
11. demonstrate and use the property that the total probability for all possible outcomes in the sample space is 1;
12. use the property that $$P(A')=1-P(A)$$, where $$P(A')$$ is the probability that event $$A$$ does not occur
13. use the property that $$P(A∪B)=P(A)+P(B)-P(A∩B)$$, for event $$A$$ and $$B$$;
14. use the property $$P(A∩B)=0$$ or $$P(A∪B)=P(A)+P(B)$$, where $$A$$ and $$B$$ are mutually exclusive events;
15. use the property $$P(A∩B)=P(A)×P(B)$$ or $$P(A|B)=P(A)$$ where $$A$$ and $$B$$ are independent events;
16. use the property $$\displaystyle P(A|B)={P(A∩B)\over P(B)}$$ where $$P(B)≠0$$.
17. use a tree diagram to list all possible outcomes for conditional probability.