1. A Survey of Probability Concepts
Pertemuan ke-5

2. Definitions
PROBABILITY (P): A value between zero and one, inclusive, describing the relative possibility (chance or likelihood) an event will occur.
The P is frequently expressed as a decimal or fraction
The P of 1 represents something that is certain to happen, and the P of 0 represents something that can not happen
The closer P is to 1, the more sure we are it will happen. The closer P is to 0, the more possible it is the event will happen.

3. Cont..
The use of probability theory allows the decision maker with only limited information to analyze the risks and minimize the gamble inherent
Three key words of probability
Experiment: The observation of some activity or the act of taking some measurement.
Outcome: A particular result of an experiment.
Event: A collection of one or more outcomes of an experiment.

4. Approaches to Probability
The objectives approaches:
Classical probability
Empirical probability/ relative frequency
The subjective approach

5. Approaches to Probability
Classical probability is based on the assumption that the outcomes of an experiment are equally likely.
It is unnecessary to do an experiment to determine probability of an event occurring
Mutually Exclusive Events: The occurrence of any one event means that none of the others can occur at the same time.
Collectively exhaustive: At least one of the events must occur when an experiment is conducted

6. Cont..
Empirical/Relative Frequency Concept : The probability of an event happening in the long run is determined by observing what fraction of the time like events happened in the past :

7. Cont..
Subjective probability: The likelihood (probability) of a particular event happening that is assigned by an individual based on whatever information is available.

8. Basic Rules of Probability
Rules of addition:
The Special rule of addition. If two events A and B are mutually exclusive, the special rule of addition states that the probability of A or B occurring equals the sum of their respective probabilities :
P(A or B) = P(A) + P(B)
The complement rule is used to determine the probability of an event occurring by subtracting the probability of the event not occurring from 1. If P(A) is the probability of event A and P(~A) is the complement of A,
P(A) + P(~A) = 1 or P(A) = 1 - P(~A).

9. Cont.. Rules of addition :
The General rule of addition. If A and B are two events that are not mutually exclusive, then P(A or B) is given by the following formula:
P(A or B) = P(A) + P(B) - P(A and B)
The Joint Probability is a probability that measures the likelihood that two or more events will happen concurrently.

10. Cont.. Rule of Multiplication :
The special rule of multiplication requires that two events A and B are independent.
Two events A and B are independent if the occurrence of one has no effect on the probability of the occurrence of the other.
P(A and B) = P(A)*P(B)

11. Cont..
Conditional probability is the probability of a particular event occurring, given that another event has occurred.
Note : The probability of the event A given that the event B has occurred is denoted by P(A|B).
The general rule of multiplication use to find the joint probability that two events will occur, one after the other
P(A and B) = P(A)*P(B/A)
where P(B/A) stand for the probability B will occur given that A has already occurred

12. Example
Survey terhadap 200 eksekutif jika ditawari posisi yang lebih baik di perusahaan lain
Berapa prob. eksekutif yang tidak pindah dan punya masa kerja lebih dari 10 tahun ?
Loyalitas
Masa Kerja
< 1 th
1 – 5 th
6 – 10 th
> 10 th
Total
Tidak Pindah 10 30 5 75
120
Pindah 25 15 80

13. Tree Diagrams
A tree diagram is very useful for portraying conditional and joint probabilities and is particularly useful for analyzing business decisions involving several stages.

14. Bayes’ Theorem
If two mutually exclusive and collectively exhaustive events :
Prior probability : the initial probability based on the present level of information
Posterior probability : a revised probability based on additional information

15. Example
Di Fakultas Ekonomi, 30% mahasiswa dan 20% mahasiswi memilih jurusan Manajemen. Diketahui 45% yang kuliah di FE adalah mahasiswi. Jika diambil satu sampel random, berapakah probabilitas yang memilih jurusan Manajemen adalah mahasiswi/berapakah probabilitas mahasiswi dengan ketentuan memilih jurusan manajemen?

17. If there are n mutually exclusive and collectively exhaustive events :

18. Example
Suatu pabrik VCR membeli microchip (LS24) dari 3 suplier dengan komposisi : Hall (30%), Schuller (20%), Crawford (50%). Berdasarkan pengalaman, 3% LS24 dari Hall, 5% dari Schuller, dan 4% dari Crawford rusak. Jika diambil satu sampel random LS24 dan ternyata rusak, berapa probabilitas LS24 tersebut berasal dari Schuller ?

20. Principles of Counting
The Multiplication Formula: If there are m ways of doing one thing and n ways of doing another thing, there are m x n ways of doing both.

21. Cont..
Permutation: Any arrangement of r objects selected from n possible objects.
Note : The order of arrangement is important in permutations.

22. Example
Terdapat 8 mesin tetapi hanya terdapat 3 ruang mesin dalam sebuah toko mesin. Berapa cara berbeda delapan mesin dapat disusun pada tiga ruangan yang tersedia?
Jawab: 8P3

23. Cont..
Combination: The number of ways to choose r objects from a group of n objects without regard to order.