PROBABILITY References: Budiyono Statistika untuk Penelitian: Edisi Kedua. Surakarta: UNS Press. Spigel, M. R Probability and Statistics. Singapore: McGraws-Hill International Book Company. Walpole, R. E Introduction to Statistics. New York: Macmillan Publishing Co.,Inc.

EXPERIMENTS, SAMPLE SPACES, AND EVENTS In statistics, we have what we call statistical experiments, or experiments, for brieftly Sample space is a set of all possible outcome of an experiment. Sample space is denoted by S The member of sample space is called sample point. An event is a subset of a sample space.

Theorem Theorem 1 On a finite sample space S, having n elements, the number of events on S is 2 n(S).

Experiment of tossing a coin once S = {A, G} or S = {H, T} n(S) = 2 The possible events on S are: 1. E 1 =  ; 2. E 2 = {A}; 3. E 3 = {G}; and 4. E 4 = {A, G} The number of events on S is 2 2

Experiment of tossing a die once S = {1, 2, 3, 4, 5, 6} n(S) = 6 The example of events on S are: 1. E 1 =  ; 2. E 2 = {1, 2, 3, 4, 5, 6}; 3. E 3 = {1, 3, 5}; and 4. E 4 = {1, 2, 4}, etc The number of events on S is 2 6

Example of tossing 3 coins once (or 1 coin three times)

S = {AAA, AAG, AGA, AGG, GAA, GAG, GGA, GGG} or S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} n(S) = 8 The number of events on S is 2 8 An example of events on S are: 1. E 1 = {} 2. E 2 = S 3. E 3 = {AAA} 4. E 4 = {AAG, AGA, GAA}, etc

Experiment of tossing 2 dice once (or 1 dice two times) S = {(1,1), (2,1), (3,1), (4,1), (5,1, (6,1), (1,2), (2,2), (3,2), (4,2), (5,2), (6,2),... (1,6), (2,6), (3,6), (4,6), (5,6), (6,6)} n(S) = 36 An example of events on S are: 1. E 1 =  ; 2. E 2 = {(1,1), (2,1), (3,1), (4,1), (5,1), (6,1)}; 3. E 3 = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}; and 4. E 4 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}, etc The number of events on S is 2 36

Experiment of tossing a die and a coin once S = {(1,A), (1,G), (2,A), (2,G), (3,A), (3,G), (4,A), (4,G), (5,A), (5,G), (6,A), (6,G)} n(S) = 12 An example of events on S are: 1. E 1 = , 2. E 2 = S 3. E 3 = {(1,A), (2,A), (3,A), (4,A), (5,A), (6,A)} 4. E 2 = {(2,A), (2,G)} 5. E 3 = {(4,A)}, etc

The Definition of Probability The definition is called classic definition or a priori definition of a probability. If an event A assosiated with an experiment having sample space S, in which all possible sample points have a same probability to occur, then the probobility of event A, denoted by P(A), is defined by the following formulae:

Example 1 Experiment of tossing a die once S = {1, 2, 3, 4, 5, 6}; n(S) = 6 The example of events on S are: 1. E 1 =  ; 2. E 2 = {1, 2, 3, 4, 5, 6}; 3. E 3 = {1, 3, 5}; and 4. E 4 = {1, 2, 4}, etc

Example 2 Experiment of tossing 2 dice once (or 1 die two times) S = {(1,1), (2,1), (3,1), (4,1), (5,1, (6,1), (1,2), (2,2), (3,2), (4,2), (5,2), (6,2),... (1,6), (2,6), (3,6), (4,6), (5,6), (6,6)} a. Find the probability of getting the same number! E 1 = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} n(E 1 ) = 6

Example 3 Experiment of tossing 2 dice once (or 1 dice two times) S = {(1,1), (2,1), (3,1), (4,1), (5,1, (6,1), (1,2), (2,2), (3,2), (4,2), (5,2), (6,2),... (1,6), (2,6), (3,6), (4,6), (5,6), (6,6)} b. Find the probability of scoring a total of 7 points E 2 = {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)} n(E 2 ) = 6

Example 4 Experiment of tossing 2 dice once (or 1 dice two times) S = {(1,1), (2,1), (3,1), (4,1), (5,1, (6,1), (1,2), (2,2), (3,2), (4,2), (5,2), (6,2),... (1,6), (2,6), (3,6), (4,6), (5,6), (6,6)} c. Find the probability of getting even score on the first die. E 3 = {(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),}

Example 5 A marble is drawn at random from a box containing 10 red, 30 white, 20 blue, and 15 orange marbles. Find the probability that it is orange or red

Example 6 A card is drawn at random from an ordinary deck of 52 playing cards. Find the probability that it is an ace

Example 7 A card is drawn at random from an ordinary deck of 52 playing cards. Find the probability that it is a jack of hearts

Example 8 A card is drawn at random from an ordinary deck of 52 playing cards. Find the probability that it is a three of clubs or a six of diamonds

The Axioms of Probability Postulate 1 P(A) is a real non-negatif number for every A in sample space S, i.e. P(A) ≥ 0 for every event A. Postulate 2 P(S) = 1 for every sample space S. Postulate 3 If A 1, A 2, A 3,... are mutually exclusive events in sample space S, then: P(A 1  A 2  A 3 ...) = P(A 1 ) + P(A 2 ) + P(A 3 ) +...

Theorems: Theorem 2 if A is an event on S and A c is the complement of A, then: P(A c ) = 1  P(A) Theorem 3 For every event A on S, 0  P(A)  1 Theorem 4 The probability of an empty set is zero, i.e. : P(  ) = 0

Theorems: Theorem 5 For every event A and B on S, P(A  B) = P(A) + P(B)  P(A  B) Theorem 6 For every event A, B, and C on S, P(A  B  C) = P(A) + P(B) + P(C)  P(A  B)  P(A  C)  P(B  C) + P(A  B  C) Theorem 7 If A and B are events on S and A  B, then: P(A)  P(B)

Example Suppose P(A) = 0.5, P(B) = 0.4, and P(A  B) = 0.3. Find: a. P(A  B) b. P(A  B c ) c. P(A c  B c ) d. P(A c  B c )

Example In a class in wich there is 40 students on it, is given the following data. As many as 25 students like football, as many as 15 students like tennis, and as many as 5 students like both sports. A person is called randomly. What is the probability that someone who called likes: a. football b. football or tennis c. football and tennis

Disjoint Events Event A and B on S are called disjoint events (kejadian saling asing, saling lepas) if: A  B = 

Conditional Probability If A and B are two events on S in which P(A)  0, then the conditional probability of B given A, denoted by P(B|A), is defined as follows:

Theorems on Conditional Probability Theorem 8 For any two events A and B on a sample space S, we have: P (A  B) = P(A) P(B|A) Theorem 9 For any events A 1, A 2, A 3,... on a a sample space S, we have: P(A 1  A 2  A 3... ) = P(A 1 ) P(A 2 |A 1 ) P(A 3 |A 1  A 2 )...

Total Probabity Theorem If events B 1, B 2, B 3,..., B k form a partition on S and P(B i )  0 for every i = 1, 2, 3,..., k, then for any event A on S in which P(A)  0, then: P(A) = P(B 1 )P(A|B 1 ) + P(B 2 )P(A|B 2 ) P(B k )P(A|B k ) It can be written as:

Example

Bayes’ Theorem Suppose events B 1, B 2, B 3,..., B k form a partition on a sample space S and P(B i )  0 for every i = 1, 2, 3,..., k. Then for any event A on S with P(A)  0, we have: for every i = 1, 2, 3,..., k

Example

The box number I contains 3 red balls and 2 blue balls. The box number II contains 2 red balls and 8 blue balls. A coin is tossed. If the “angka” occurs, then any ball is drawn randomly from the box number I. Otherwise, if the “gambar” occurs then any ball is drawn randomly from the box number II. a. What is the probability that the ball is drawn from the box number I? b. What is the probability that the ball is drawn from the box number II? c. What is the probability any red ball is drawn? d. What is the probability any blue ball is drawn? e. Someone who toss the coin does not know wether occurs “angka” or “gambar”, but he knows that a red ball is drawn. What is the probability that the ball is drawn from the box number I?

Independent Events Definition: Any two events A and B are called independent events if P(A  B) = P(A)P(B) Theorems: If A and B are independent events, then: 1. A and B c are independent events, 2. A c and B are independent events, and 3. A c and B c are independent events

Example Three balls are drawn successively from a box containing 6 red balls, 4 white balls, and 5 blue balls. Find the probability that they are drawn in order red, white, and blue, if the sampling is: a. with replacement b. without replacement

Example Given two independent events A and B having P(A) = 0.5 and P(B) = 0.4. Find: