38 2 Some Probabilistic Concepts and Results Then, by Theorem 6, there are 13 ·6 ·220·64 = 1,098,240 poker hands with one pair. Hence, by assuming the uniform probability measure, the required probability is equal to 1 098 240 2 598 960 042 ,, ,, ≈ i) The number of ways in which n distinct balls can be distributed into k distinct cells is k n . ii) The number of ways that n distinct balls can be distributed into k distinct cells so that the jth cell contains n j balls (n j ≥ 0, j = 1, . . . , k, Σ k j=1 n j = n) is n nn n n nn n k k ! !! ! ,, , . 12 12 ⋅⋅⋅ = ⋅⋅⋅ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ iii) The number of ways that n indistinguishable balls can be distributed into k distinct cells is kn n +− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 . Furthermore, if n ≥ k and no cell is to be empty, this number becomes n k − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 1 . PROOF i) Obvious, since there are k places to put each of the n balls. ii) This problem is equivalent to partitioning the n balls into k groups, where the jth group contains exactly n j balls with n j as above. This can be done in the following number of ways: n n nn n nn n n n nn n k k k 1 1 2 11 12 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⋅⋅⋅ − −⋅⋅⋅− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⋅⋅⋅ − ! !! ! . iii) We represent the k cells by the k spaces between k + 1 vertical bars and the n balls by n stars. By fixing the two extreme bars, we are left with k + n − 1 bars and stars which we may consider as k + n − 1 spaces to be filled in by a bar or a star. Then the problem is that of selecting n spaces for the n stars which can be done in kn n +− () 1 ways. As for the second part, we now have the condition that there should not be two adjacent bars. The n stars create n − 1 spaces and by selecting k − 1 of them in n k − − () 1 1 ways to place the k − 1 bars, the result follows. ▲ THEOREM 9 2.4 Combinatorial Results 39 REMARK 5 i) The numbers n j , j = 1, , k in the second part of the theorem are called occupancy numbers. ii) The answer to (ii) is also the answer to the following different question: Consider n numbered balls such that n j are identical among themselves and distinct from all others, n j ≥ 0, j = 1, , k, Σ k j=1 n j = n. Then the number of different permutations is n nn n k12 ,, , . ⋅⋅⋅ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Now consider the following examples for the purpose of illustrating the theorem. Find the probability that, in dealing a bridge hand, each player receives one ace. The number of possible bridge hands is N = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = () 52 52 13 4 13, 13, 13, 13 ! ! . Our sample space S is a set with N elements and assign the uniform probability measure. Next, the number of sample points for which each player, North, South, East and West, has one ace can be found as follows: a) Deal the four aces, one to each player. This can be done in 4 1 4 4 , ! ! 1, 1, 1 1! 1! 1! 1! ways. ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ == b) Deal the remaining 48 cards, 12 to each player. This can be done in 48 12 48 4 , ! 12, 12, 12 12! ways. ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = () Thus the required number is 4!48!/(12!) 4 and the desired probability is 4!48!(13!) 4 /[(12!) 4 (52!)]. Furthermore, it can be seen that this probability lies between 0.10 and 0.11. The eleven letters of the word MISSISSIPPI are scrambled and then arranged in some order. i) What is the probability that the four I’s are consecutive letters in the resulting arrangement? There are eight possible positions for the first I and the remaining seven letters can be arranged in 7 142,, () distinct ways. Thus the required probability is EXAMPLE 8 EXAMPLE 7 40 2 Some Probabilistic Concepts and Results 8 7 4 165 002 1, 4, 2 11 1, 4, 4, 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =≈ ii) What is the conditional probability that the four I’s are consecutive (event A), given B, where B is the event that the arrangement starts with M and ends with S? Since there are only six positions for the first I, we clearly have PAB () = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =≈ 6 5 2 9 4 1 21 005 , 3, 2 iii) What is the conditional probability of A, as defined above, given C, where C is the event that the arrangement ends with four consecutive S’s? Since there are only four positions for the first I, it is clear that PAC () = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =≈ 4 3 2 7 1 4 35 011 , 2, 4 Exercises 2.4.1 A combination lock can be unlocked by switching it to the left and stopping at digit a, then switching it to the right and stopping at digit b and, finally, switching it to the left and stopping at digit c. If the distinct digits a, b and c are chosen from among the numbers 0, 1, . . . , 9, what is the number of possible combinations? 2.4.2 How many distinct groups of n symbols in a row can be formed, if each symbol is either a dot or a dash? 2.4.3 How many different three-digit numbers can be formed by using the numbers 0, 1, . . . , 9? 2.4.4 Telephone numbers consist of seven digits, three of which are grouped together, and the remaining four are also grouped together. How many num- bers can be formed if: i) No restrictions are imposed? ii) If the first three numbers are required to be 752? 2.4 Combinatorial Results 41 2.4.5 A certain state uses five symbols for automobile license plates such that the first two are letters and the last three numbers. How many license plates can be made, if: i) All letters and numbers may be used? ii) No two letters may be the same? 2.4.6 Suppose that the letters C, E, F, F, I and O are written on six chips and placed into an urn. Then the six chips are mixed and drawn one by one without replacement. What is the probability that the word “OFFICE” is formed? 2.4.7 The 24 volumes of the Encyclopaedia Britannica are arranged on a shelf. What is the probability that: i) All 24 volumes appear in ascending order? ii) All 24 volumes appear in ascending order, given that volumes 14 and 15 appeared in ascending order and that volumes 1–13 precede volume 14? 2.4.8 If n countries exchange ambassadors, how many ambassadors are involved? 2.4.9 From among n eligible draftees, m men are to be drafted so that all possible combinations are equally likely to be chosen. What is the probability that a specified man is not drafted? 2.4.10 Show that n m n m n m + + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = + + 1 1 1 1 . 2.4.11 Consider five line segments of length 1, 3, 5, 7 and 9 and choose three of them at random. What is the probability that a triangle can be formed by using these three chosen line segments? 2.4.12 From 10 positive and 6 negative numbers, 3 numbers are chosen at random and without repetitions. What is the probability that their product is a negative number? 2.4.13 In how many ways can a committee of 2n + 1 people be seated along one side of a table, if the chairman must sit in the middle? 2.4.14 Each of the 2n members of a committee flips a fair coin in deciding whether or not to attend a meeting of the committee; a committee member attends the meeting if an H appears. What is the probability that a majority will show up in the meeting? 2.4.15 If the probability that a coin falls H is p (0 < p < 1), what is the probability that two people obtain the same number of H’s, if each one of them tosses the coin independently n times? Exercises 41 42 2 Some Probabilistic Concepts and Results 2.4.16 i) Six fair dice are tossed once. What is the probability that all six faces appear? ii) Seven fair dice are tossed once. What is the probability that every face appears at least once? 2.4.17 A shipment of 2,000 light bulbs contains 200 defective items and 1,800 good items. Five hundred bulbs are chosen at random, are tested and the entire shipment is rejected if more than 25 bulbs from among those tested are found to be defective. What is the probability that the shipment will be accepted? 2.4.18 Show that M m M m M m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 11 1 , where N, m are positive integers and m < M. 2.4.19 Show that m x n rx mn r x r ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ∑ 0 , where k x xk ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =>0if . 2.4.20 Show that i n j ii n j n j j n j n );) . ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− () ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = == ∑∑ 210 00 2.4.21 A student is given a test consisting of 30 questions. For each question there are supplied 5 different answers (of which only one is correct). The student is required to answer correctly at least 25 questions in order to pass the test. If he knows the right answers to the first 20 questions and chooses an answer to the remaining questions at random and independently of each other, what is the probability that he will pass the test? 2.4.22 A student committee of 12 people is to be formed from among 100 freshmen (60 male + 40 female), 80 sophomores (50 male + 30 female), 70 juniors (46 male + 24 female), and 40 seniors (28 male + 12 female). Find the total number of different committees which can be formed under each one of the following requirements: i) No restrictions are imposed on the formation of the committee; ii) Seven students are male and five female; 2.4 Combinatorial Results 43 iii) The committee contains the same number of students from each class; iv) The committee contains two male students and one female student from each class; v) The committee chairman is required to be a senior; vi) The committee chairman is required to be both a senior and male; vii) The chairman, the secretary and the treasurer of the committee are all required to belong to different classes. 2.4.23 Refer to Exercise 2.4.22 and suppose that the committee is formed by choosing its members at random. Compute the probability that the committee to be chosen satisfies each one of the requirements (i)–(vii). 2.4.24 A fair die is rolled independently until all faces appear at least once. What is the probability that this happens on the 20th throw? 2.4.25 Twenty letters addressed to 20 different addresses are placed at ran- dom into the 20 envelopes. What is the probability that: i) All 20 letters go into the right envelopes? ii) Exactly 19 letters go into the right envelopes? iii) Exactly 17 letters go into the right envelopes? 2.4.26 Suppose that each one of the 365 days of a year is equally likely to be the birthday of each one of a given group of 73 people. What is the probability that: i) Forty people have the same birthdays and the other 33 also have the same birthday (which is different from that of the previous group)? ii) If a year is divided into five 73-day specified intervals, what is the probabil- ity that the birthday of: 17 people falls into the first such interval, 23 into the second, 15 into the third, 10 into the fourth and 8 into the fifth interval? 2.4.27 Suppose that each one of n sticks is broken into one long and one short part. Two parts are chosen at random. What is the probability that: i) One part is long and one is short? ii) Both parts are either long or short? The 2n parts are arranged at random into n pairs from which new sticks are formed. Find the probability that: iii) The parts are joined in the original order; iv) All long parts are paired with short parts. 2.4.28 Derive the third part of Theorem 9 from Theorem 8(ii). 2.4.29 Three cards are drawn at random and with replacement from a stan- dard deck of 52 playing cards. Compute the probabilities P(A j ), j = 1, . . . , 5, where the events A j , j = 1, . . . , 5 are defined as follows: Exercises 43 44 2 Some Probabilistic Concepts and Results As s As s As s As s As s 1 2 3 4 5 1 =∈ {} =∈ {} =∈ {} =∈ { } =∈ {} S S S S S ;, ;, ;, ; , ;. all 3 cards in are black at least 2 cards in are red exactly 1 card in is an ace the first card in is a diamond, the second is a heart and the third is a club card in is a diamond, 1 is a heart and 1 is a club 2.4.30 Refer to Exercise 2.4.29 and compute the probabilities P(A j ), j = 1, . . . , 5 when the cards are drawn at random but without replacement. 2.4.31 Consider hands of 5 cards from a standard deck of 52 playing cards. Find the number of all 5-card hands which satisfy one of the following requirements: i) Exactly three cards are of one color; ii) Three cards are of three suits and the other two of the remaining suit; iii) At least two of the cards are aces; iv) Two cards are aces, one is a king, one is a queen and one is a jack; v) All five cards are of the same suit. 2.4.32 An urn contains n R red balls, n B black balls and n W white balls. r balls are chosen at random and with replacement. Find the probability that: i) All r balls are red; ii) At least one ball is red; iii) r 1 balls are red, r 2 balls are black and r 3 balls are white (r 1 + r 2 + r 3 = r); iv) There are balls of all three colors. 2.4.33 Refer to Exercise 2.4.32 and discuss the questions (i)–(iii) for r = 3 and r 1 = r 2 = r 3 (= 1), if the balls are drawn at random but without replacement. 2.4.34 Suppose that all 13-card hands are equally likely when a standard deck of 52 playing cards is dealt to 4 people. Compute the probabilities P(A j ), j = 1, . . . , 8, where the events A j , j = 1, . . . , 8 are defined as follows: As s As s As s As s As s As s 1 2 3 4 5 6 =∈ {} =∈ {} =∈ {} =∈ {} =∈ {} =∈ {} S S S S S S ;, ;, ;, ;, ;, ;, consists of 1 color cards consists only of diamonds consists of 5 diamonds, 3 hearts, 2 clubs and 3 spades consists of cards of exactly 2 suits contains at least 2 aces does not contain aces, tens and jacks 2.4 Combinatorial Results 452.5 Product Probability Spaces 45 As s As s 7 8 =∈ {} =∈ {} S S ;, ;. consists of 3 aces, 2 kings and exactly 7 red cards consists of cards of all different denominations 2.4.35 Refer to Exercise 2.4.34 and for j = 0, 1, . . . , 4, define the events A j and also A as follows: As s j As s j =∈ {} =∈ {} S S ;, ;. contains exactly tens contains exactly 7 red cards For j = 0, 1, . . . , 4, compute the probabilities P(A j ), P(A j |A) and also P(A); compare the numbers P(A j ), P(A j |A). 2.4.36 Let S be the set of all n 3 3-letter words of a language and let P be the equally likely probability function on the events of S. Define the events A, B and C as follows: As s Bs s A Cs s =∈ {} =∈ () { } ∈ {} S S S ;, ; , . begins with a specific letter has the specified letter mentioned in the definition of in the middle entry = ; has exactly two of its letters the same Then show that: i) P(A ∩ B) = P(A)P(B); ii) P(A ∩ C) = P(A)P(C); iii) P(B ∩ C) = P(B)P(C); iv) P(A ∩ B ∩ C) ≠ P(A)P(B)P(C). Thus the events A, B, C are pairwise independent but not mutually independent. 2.5* Product Probability Spaces The concepts discussed in Section 2.3 can be stated precisely by utilizing more technical language. Thus, if we consider the experiments E 1 and E 2 with re- spective probability spaces (S 1 , A 1 , P 1 ) and (S 2 , A 2 , P 2 ), then the compound experiment (E 1 , E 2 ) = E 1 × E 2 has sample space S = S 1 × S 2 as defined earlier. The appropriate σ -field A of events in S is defined as follows: First define the class C by: CAA=× ∈ ∈ {} ×= () ∈∈ {} AAA A AA ss sAs A 121 12 2 12 12112 2 ;, , ,; , .where 46 2 Some Probabilistic Concepts and Results Then A is taken to be the σ -field generated by C (see Theorem 4 in Chapter 1). Next, define on C the set function P by P(A 1 × A 2 ) = P 1 (A 1 )P 2 (A 2 ). It can be shown that P determines uniquely a probability measure on A (by means of the so-called Carathéodory extension theorem). This probability measure is usually denoted by P 1 × P 2 and is called the product probability measure (with factors P 1 and P 2 ), and the probability space (S, A, P) is called the product probability space (with factors (S j , A j , P j ), j = 1, 2). It is to be noted that events which refer to E 1 alone are of the form B 1 = A 1 × S 2 , A 1 ∈ A 1 , and those referring to E 2 alone are of the form B 2 = S 1 ×A 2 , A 2 ∈ A 2 . The experiments E 1 and E 2 are then said to be independent if P(B 1 ∩ B 2 ) = P(B 1 )P(B 2 ) for all events B 1 and B 2 as defined above. For n experiments E j , j = 1, 2, . . . , n with corresponding probability spaces (S j , A j , P j ), the compound experiment (E 1 , , E n ) = E 1 × ···× E n has prob- ability space (S, A, P), where SS S S= ×⋅⋅⋅× = ⋅⋅⋅ () ∈= ⋅⋅⋅ {} 11 1 nnjj sssj n,,; , , ,, 2, A is the σ -field generated by the class C, where CA= ×⋅⋅⋅× ∈ = ⋅⋅⋅ {} AAAj n nj j1 1; 2, ,, ,, and P is the unique probability measure defined on A through the relationships PA A PA PA A j n nnjj11 1×⋅⋅⋅× () = () ⋅⋅⋅ () ∈= ⋅⋅⋅ ,,,, 2, .A The probability measure P is usually denoted by P 1 × ···× P n and is called the product probability measure (with factors P j , j = 1, 2, . . . , n), and the probabil- ity space (S, A, P) is called the product probability space (with factors (S j , A j , P j ), j = 1, 2, . . . , n). Then the experiments E j , j = 1, 2, . . . , n are said to be independent if P(B 1 ∩ ···∩ B 2 ) = P(B 1 )···P(B 2 ), where B j is defined by BAjn jjjjn = ×⋅⋅⋅× × × ×⋅⋅⋅× = ⋅⋅⋅ −+ SS S S 111 1,, ,. 2, The definition of independent events carries over to σ -fields as follows. Let A 1 , A 2 be two sub- σ -fields of A. We say that A 1 , A 2 are independent if P(A 1 ∩ A 2 ) = P(A 1 )P(A 2 ) for any A 1 ∈ A 1 , A 2 ∈ A 2 . More generally, the σ -fields A j , j = 1, 2, . . . , n (sub- σ -fields of A) are said to be independent if PA PA A j n j j n j j n jj = = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = () ∈= ⋅⋅⋅ ∏ 1 1 1 I for any 2, A ,, ,. Of course, σ -fields which are not independent are said to be dependent. At this point, notice that the factor σ -fields A j , j = 1, 2, . . . , n may be considered as sub- σ -fields of the product σ -field A by identifying A j with B j , where the B j ’s are defined above. Then independence of the experiments E j , j = 1, 2, . . . , n amounts to independence of the corresponding σ -fields A j , j = 1, 2, . . . , n (looked upon as sub- σ -fields of the product σ -field A). 2.4 Combinatorial Results 47 Exercises 2.5.1 Form the Cartesian products A × B, A × C, B × C, A × B × C, where A = {stop, go}, B = {good, defective), C = {(1, 1), (1, 2), (2, 2)}. 2.5.2 Show that A × B =∅ if and only if at least one of the sets A, B is ∅. 2.5.3 If A ⊆ B, show that A × C ⊆ B × C for any set C. 2.5.4 Show that i) (A × B) c = (A × B c ) + (A c × B) + (A c × B c ); ii) (A × B) ∩ (C × D) = (A ∩ C) × (B ∩ D); iii) (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D) − [(A ∩ C c ) × (B c ∩ D) + (A c ∩ C) × (B ∩ D c )]. 2.6* The Probability of Matchings In this section, an important result, Theorem 10, is established providing an expression for the probability of occurrence of exactly m events out of possible M events. The theorem is then illustrated by means of two interesting exam- ples. For this purpose, some additional notation is needed which we proceed to introduce. Consider M events A j , j = 1, 2, . . . , M and set S SPA SPAA SPAAA SPAA A j j M jj jjM rjjj jj jM MM r r 0 1 1 2 1 1 12 1 12 12 12 12 = = () =∩ () = ∩ ∩⋅⋅⋅∩ () = ∩ ∩⋅⋅⋅∩ () = ≤< ≤ ≤ < <⋅⋅⋅< ≤ ∑ ∑ ∑ , , , , . M M Let also B C D mAjM m m m j = = = ⎫ ⎬ ⎪ ⎭ ⎪ = ⋅⋅⋅ exactly at least at most of the events , 2, occur.,,1 Then we have 2.6* The Probability of Matchings 47 [...]... X? 3 .2. 2 It has been observed that 12. 5% of the applicants fail in a certain screening test If X stands for the number of those out of 25 applicants who fail to pass the test, what is the probability that: 62 3 On Random Variables and Their Distributions iii) X ≥ 1? iii) X ≤ 20 ? iii) 5 ≤ X ≤ 20 ? 3 .2. 3 A manufacturing process produces certain articles such that the probability of each article being defective... = ( ) P A ( ) ( ) P A +P B , as asserted ▲ It is possible to interpret B as a catastrophic event, and A as an event consisting of taking certain precautionary and protective actions upon the energizing of a signaling device Then the significance of the above probability becomes apparent As a concrete illustration, consider the following simple example (see also Exercise 2. 6.3)  52 2 Some Probabilistic... Some General Concepts Given a probability space (S, class of events, P), the main objective of probability theory is that of calculating probabilities of events which may be of importance to us Such calculations are facilitated by a transformation of the sample space S, which may be quite an abstract set, into a subset of the real line ‫ ޒ‬with which we are already familiar This is, actually, achieved... results EXAMPLE 10 Coupon collecting (case of sampling with replacement) Suppose that a manufacturer gives away in packages of his product certain items (which we take to be coupons), each bearing one of the integers 1 to M, in such a way that each of the M items is equally likely to be found in any package purchased If n packages are bought, show that the probability that exactly m of the integers,... balls among the r balls selected, and f(x) is the probability that this number is exactly x Here S = {all r-sequences of R’s and B’s}, where R stands for a red ball and B stands for a black ball The urn/balls model just described is a generic model for situations often occurring in practice For instance, the urn and the balls may be replaced by a box containing certain items manufactured by a certain... Continuous Random Variables (and General Concepts 3.1 Soem Random Vectors) 65 3 .2. 22 (Polya’s urn scheme) Consider an urn containing b black balls and r red balls One ball is drawn at random, is replaced and c balls of the same color as the one drawn are placed into the urn Suppose that this experiment is repeated independently n times and let X be the r.v denoting the number of black balls drawn Then... several values of σ I2 = 1 2 ∞ ∫ ∫ ∞ −∞ −∞ e ( − z 2 +υ 2 ) 2 dz dν = 1 ∞ 2 ∫ ∫ 0 2 0 e −r 2 2 r dr dθ by the standard transformation to polar coordinates Or I2 = 1 2 ∫ ∞ 0 e −r 2 2 r dr ∫ 2 0 ∞ dθ = ∫ e − r 2 2 0 r dr = − e − r 2 2 ∞ 0 = 1; that is, I2 = 1 and hence I = 1, since f(x) > 0 It is easily seen that f(x) is symmetric about x = μ, that is, f(μ − x) = f(μ + x) and that f(x) attains its maximum... or 10, the player continues rolling the dice until either the same sum appears before a sum of 7 appears in which case he wins, or until a sum of 7 appears before the original sum appears in which case the player loses It is assumed that the game terminates the first time the player wins or loses What is the probability of winning? 3.1 Soem General Concepts 53 Chapter 3 On Random Variables and Their Distributions... is possible Again, when one is presented with a function f and is asked whether f is a p.d.f (of some r vector), all one has to check is non-negativity of f, and that the sum of its values or its integral (over the appropriate space) is equal to 1 3 .2 Discrete Random Variables (and Random Vectors) 3 .2. 1 Binomial The Binomial distribution is associated with a Binomial experiment; that is, an experiment... ∩ A2 + A1 c ∩ B1c ∩ A2 ∩ B2 ∩ A3 ( 51 ] ) ) c c + ⋅ ⋅ ⋅ + A1 c ∩ B1c ∩ ⋅ ⋅ ⋅ ∩ An ∩ Bn ∩ An+1 + ⋅ ⋅ ⋅ ( ) ( ) ( c c = P A1 + P A1 c ∩ B1c ∩ A2 + P A1 c ∩ B1c ∩ A2 ∩ B2 ∩ A3 ( ) ) = P( A ) + P( A ∩ B ) P( A ) + P( A ∩ B ) P( A ∩ B ) P( A ) + ⋅ ⋅ ⋅ + P( A ∩ B ) ⋅ ⋅ ⋅ P( A ∩ B )P( A ) + ⋅ ⋅ ⋅ (by Theorem 6) = P( A) + P( A ∩ B )P( A) + P ( A ∩ B )P( A) + ⋅ ⋅ ⋅ + P ( A ∩ B )P( A) ⋅ ⋅ ⋅ = P( A) [1 + P( A ∩ B ) . s 1 2 3 4 5 1 =∈ {} =∈ {} =∈ {} =∈ { } =∈ {} S S S S S ;, ;, ;, ; , ;. all 3 cards in are black at least 2 cards in are red exactly 1 card in is an ace the first card in is a diamond, the second is a heart and the third is a club card in is a diamond, 1 is a. events carries over to σ -fields as follows. Let A 1 , A 2 be two sub- σ -fields of A. We say that A 1 , A 2 are independent if P (A 1 ∩ A 2 ) = P (A 1 )P (A 2 ) for any A 1 ∈ A 1 , A 2 ∈ A 2 . More. suit. 2. 4. 32 An urn contains n R red balls, n B black balls and n W white balls. r balls are chosen at random and with replacement. Find the probability that: i) All r balls are red; ii) At least