Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated. c. With the calculated measures, statistical inferences are conducted. Why are statistical inferences required? Example) people in group A are taller than people in group B? A (cm) B (cm) 174 164 183 175 177 173 185 166-1 -
0 20 40 60 80 100 130 140 150 160 170 180 190 200 130 140 150 160 170 180 190 200 130 140 150 160 170 180 190 200 Most experiments cannot be predicted with certainty!!! - 2 -
1) Definition Outcome : the result from each experiment Random experiment : experiment that can be repeated under the same conditions Sample (outcome) space : collection of every possible outcome, denoted by Example 1.1.1 Two dice are cast, and the total number of spots on the sides that are "up" are counted. What is the outcome space? Example 1.1.2 A fair coin is flipped successively at random until the first head is observed. If we let x denote the number of flips of the coin that are required, then S = { x: x = 1, 2,..., }. Example 1.1.3 In the cast of one red die and one white die, let the outcome be the ordered pair - 3 -
2) Probability (traditional definition) Event : subset of sample space ; event A has occurred outcome of the experiment is in A Let A be a part of collection of outcomes in S; that is, A S. When the random experiment is performed and the outcome of the experiment is in A, we say that event A has occurred. Relative frequency (RF): In n repetitions of the random experiment, relative frequency of the event A = where is the number of times that the event A actually occurred. As n increases, In future performances of the experiment, the RF of the event A will either equal or approximate The is called the probability that the outcome of the random experiment is in A and denoted by P(A). - 4 -
Example 1.1.4 (Ex. 1.1-3 continued) the event that the sum of the pair is equal to seven Suppose:, the RF of the event = Thus, a number, that is close to 0.15, would be called the probability of the event (sol) It converges to 6/36 0.166667 Example 1.1.5 A fair six-sided die is rolled six times. If the face numbered k is the outcome on roll k for k = 1, 2,..., 6, we say that a match has occurred. The experiment is called a success if at least one match occurs during the six trials. n N(A) N(A) / n 50 37 0.740 100 69 0.690 250 172 0.688 500 330 0.660 (sol) It converges to 1 - (5/6) 6 0.665102-5 -
Fraction of experiments having at least one match RE 0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500 n - 6 -
1.2 Properties of Probability 1) Properties of set Subset, denoted by if for. if and Example 1.2.1, Example 1.2.2, Null set : null set if a set has no elements Union of sets or or or or for some - 7 -
Example 1.2.3 or, or Example 1.2.4 (sol) (i) For any or, for positive integer (ii) For, there are s.t.. Therefore, By (i) and (ii), Intersection of sets and - 8 -
for all Example 1.2.5, Example 1.2.6, Example 1.2.7 (sol) (i) For any or, for positive integer - 9 -
(ii) For, there are s.t.. Therefore, By (i) and (ii), Elementary algebra of sets a), Example 1.2.8, b), c), Complement: = where : space - 10 -
Note : Example 1.2.9 In tossing a coin four times, and if,. Properties of complement a), b), c) (sol) (a) ( ) Any event is a subset of sample space. Therefore, ( ) For any, either or by the definition of complement. Thus, (sol) (c) - 11 -
( ) For any,. Thus, it is proven. ( ) For any,. Thus, it is proven. Mutually exclusive events are mutually exclusive events if for. Exhaustive events are exhaustive events if. DeMorgan's Laws Suppose : space and,, (sol) (a) (i) For, we have and - 12 -
or or (ii) For, by definition of union or or and (sol) (b) because (i) If we let and, we have because of (a). (ii) - 13 -
2) Definition of Probability Set function A function that assigns real value to each event (set). For instance, if we define f(a) indicates the number of the elements in the set A, f(a) is a set function. Example f(a) = the number of elements in A. f(a) is a set function. f(a) = the maximum of elements in A. f(a) is a set function. - 14 -
-field: Let be a collection of subsets of. We say is a -field if (1) (2) If, then (3) If, then Example For any event, is a -field. Example (i) In tossing a coin,. Confirm whether is a -field. (ii) In casting a die,. Confirm whether is a -field. (sol) (i) It is a -field. (ii) It is not a -field. For instance, - 15 -
Probability set function : advanced definition of Probability Suppose : sample space, and : a real valued function defined on on -field,. (1) S (2) (3) (Countable Additivity) If for, If satisfies the three conditions (1), (2) and (3), is called a probability (set) function. The traditional definition of probability is also probability set function because, and for mutually exclusive event A and B. There are many probability set functions. For example, let S = {1, 2,..., 6} 1 Let P 1 (A) = # of elements in A 1/6 2 Let P 2 (A) = 1 if 1 A and otherwise P 2 (A) = 0. 3 Let P 3 (A) = 1 if there is at least a single even number in A and otherwise P 3 (A) =0. - 16 -
Are P 1 ( ), P 2 ( ) and P 3 ( ) probability set functions? P 1 ( ), P 2 ( ): Yes P 3 ( ): No because of countable additivity. For instance, P 3 ({1,2,3,4}) = 1, P 3 ({1,2}) = 1, P 3 ({3,4}) = 1. To be a probability set function, P 3 ({1,2,3,4}) has to be equal to P 3 ({1,2}) + P 3 ({3,4}) = 2. But it is not satisfied. Rules of Probability a) (Proof) : mutually exclusive and exhaustive Thus b) (Proof) : mutually exclusive and exhaustive Thus c) (Proof) : mutually exclusive Thus d) - 17 -
(Proof) and from (c), we can have that e) (Proof), : mutually exclusive, and, : mutually exclusive Thus (Proof) - 18 -
At the same time, because,. As a result, we have (proof) Mathematical induction We assume that the equation is satisfied when n k. When n = k + 1, - 19 -
By assumption, - 20 -
Thus, (i) (ii) (iii) (iv) - 21 -
(v) Therefore, f) Suppose. Then we have lim (Proof) for Then, D i : mutually exclusive and.. - 22 -
lim lim lim *** Is lim always true?? Suppose S = {0, 1} and P(0) = P(1) = 0.5. If we define C i = {0} for even number i and C i = {1} for odd number i, then what happens? g) Suppose. Then we have lim (Proof) Let Then, and the property (f) can be applied as follows: lim (LHS) : lim lim lim lim - 23 -
(RHS): As a result, we have lim lim We need to prove (i) ( ) For any, for some for some by definition of complement. for all (ii) ( ) - 24 -
For any, for all by definition. for some for some h) (Proof). By the property (f), we can have because. lim - 25 -
lim lim lim lim - 26 -
1.3 Methods of Enumeration Multiplication principle Suppose that an experiment E 1 had n 1 outcomes and for each of these possible outcomes, an experiment E 2 has n 2 possible outcomes. Then composite experiment E 1 E 2 that consists of performing first E 1 and then E 2 has n 1 n 2 possible outcomes. <Tree diagram> Permutation Each of the arrangements of n different objects is called a permutation of the n object. Sampling with replacement ( 복원추출 ) - 27 -
It occurs when an object is selected and then replaced before the next object is selected. Sampling without replacement ( 비복원추출 ) It occurs when an object is not replaced after it has been selected. Example 1.3.1 If only r positions are to be filled with objects selected from n different objects, then the number of possible ordered arrangements is Sol) n! = n P r (n r)! => Example 1.3.2 The number of ways in which r objects can be selected without replacements from n objects when the order of selection is disregarded is. Suppose that a set contains n objects of two types: r of one type and n-r of the other type. Then, the number of distinguishable arrangements is also. Sol) i) n! = n P r (n r)! = n C r r! (n r)! - 28 -
ii) n! = n C r (n r)! r! Distinguishable permutation Each of the permutations of n objects, r of one type and n-r of another type. Example 1.3.3 1 What is the number of possible 5-card hands (in 5-card poker) drawn from a deck of 52 playing cards? 2 When five cards are drawn from a deck of 52 playing cards, what is the number of outcomes in which exactly 3 cards are kings and exactly two cards are queens (event A)? 3 When five cards are drawn from a deck of 52 playing cards, what is the number of outcomes in which there are exactly two kings, two queens, and one jack (event B)? 4 If we assume that each of the five-card hands drawn from a deck of 52 playing cards has the same probability of being selected, then what are the probabilities of A and B respectively? Sol) 1 52 C 5 =2,598,960 2 4 C 3 4 C 2, 3 4 C 2 4 C 2 4 C 1, 4 4 C 3 4 C 2 / 52 C 5, 4 C 2 4 C 2 4 C 1 / 52 C 5 Example 1.3.2 Suppose that in a set of n objects, n 1 are similar, n 2 are similar,..., n s are similar, where n 1 + n 2 +... + n s = n. Then the number of distinguishable permutations of the n objects is - 29 -
(Sol). Example 1.3.3 Among nine orchids for a line of orchids along one wall, there are three white, four lavender, and two yellow. Then what is the number of different color displays? Sol) Example 1.3.4 We have the following expansion with combination: - 30 -
1.4 Conditional Probability conditional probability The conditional probability of the event, given the event, provided. multiplication rule partition : partition of C 's are mutually exclusive and law of total probability If : partition of, and, - 31 -
. Here, is called prior probability, and posterior probability Sol) Example 1.4.1 A hands of 5 cards are drawn without replacement from a deck of 52 playing cards. Calculate the conditional probability of an all spade hand ( ), given that there are at least 4 spades in the hand ( ). Sol) Example 1.4.2 A bowl contains eight chips: three RED chips and the remaining five are BLUE. Two chips are to be drawn successively w/o replacement. Compute the probability that the first draw results in a red chip ( ) and that the second draw results in a blue chip ( ). Sol) - 32 -
Example 1.4.3 From an ordinary deck of playing cards, cards are to be drawn successively, at random and w/o replacement. Compute the probability that the third spade appears on the sixth draw. : the event of two spades in the first five draws : the event of a spade on the sixth draw. Sol) Example 1.4.4 The conditional probability satisfies the axioms for a probability function, namely P(B) > 0. Sol) For any A B i) P(A B) 0, ii) P(S B) = P(B B) = 1 iii) If A 1,..., A i,... : mutually exclusive, - 33 -
because and for and for - 34 -
1.5 Independence independence of two events Two events, are called independent if and only if. Otherwise they are called dependent events. :independent when when :independent :independent Sol) iii) ( ) ( ) - 35 -
Independence of many events a) pairwise independence : pairwise independent b) mutual independence : mutually independent,, *** Mutually independent pairwise independent but the converse does not work. Independent experiment a sequence of experiments in such a way that the events associated with one of them are independent of the events associated with the others. e.g., independent flips of a coin, independent casts of a die Example 1.5.1 If A, B, and C are mutually independent events, then the following events are also independent: (a) A and (B C) - 36 -
(b) A and (B C) (c) A c and (B C c ) Sol) b) c) Example 1.5.2 Suppose that on five consecutive days an "instant winner" lottery ticket is purchased and the probability of winning is 1/5 on each day. Assuming independent trials, what is the probability of purchasing two winning tickets and three losing tickets? Sol) - 37 -
1.6 Bayes's theorem Bayes's theorem If : partition of, and,. Example 1.6.1 Bowl : 3 red and 7 blue chips and bowl : 8 red and 2 blue chips. A die is cast and bowl is selected if five or six spots show up; otherwise, bowl is selected. Compute the probability that the conditional probability of bowl, given that a red chip is drawn. Sol) Example 1.6.2 In a certain factory, machines I, II and III are all producing springs of the same length. Of their production, machines I, II, III respectively produce 2%, 1% and 3% defective springs. Of the total production of springs in the factory, machine I produces 35%, machine II produces 25% and - 38 -
machine III produces 40%. If one spring is selected at random from the total springs produced in a day, what is the probability that it is defective? If the selected spring is defective, what is the conditional probability that it was produced by machine III? Sol) - 39 -