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elementary probability for applications

Questions and Answers of Elementary Probability For Applications

15. Two identical decks of cards, each containing N cards, are shuffled randomly. We say that a k-matching occurs if the two decks agree in exactly k places. Show that the probability that there is a
The following morning, the porter was rebuked by the Bursar, so that in the evening she was careful to hang only one key on each hook. But she still only managed to hang them independently and at
(b) One evening, a bemused lodge-porter tried to hang n keys on their n hooks, but only managed to hang them independently and at random. There was no limit to the number of keys which could be hung
Find two other relations for sn and mn in terms of cn−1, sn−1, and mn−1, and hence find cn, sn, and mn. (Oxford 1974M)14. (a) Let P(A) denote the probability of the occurrence of an event A.
Let cn be the probability that a particular corner site is occupied after n such independent moves, and let the corresponding probabilities for an intermediate site at the side of the board and for a
13. A square board is divided into 16 equal squares by lines drawn parallel to its sides. A counter is placed at random on one of these squares and is then moved n times. At each of these moves, it
Use this difference equation to show that u8 = 208 256 .* 12. Any number ω ∈ [0, 1] has a decimal expansionω = 0.x1x2 . . . , and we write fk (ω, n) for the proportion of times that the integer
11. Show that if un is the probability that n tosses of a fair coin contain no run of 4 heads, then for n ≥ 4 un = 1 2un−1 + 1 4 un−2 + 1 8 un−3 + 1 16un−4.
10. In the circuits in Figure 1.2, each switch is closed with probability p, independently of all other switches. For each circuit, find the probability that a flow of current is possible between A
Fig. 1.2 Two electrical circuits incorporating switches.
9. Two people toss a fair coin n times each. Show that the probability they throw equal numbers of heads is 2n n1 22n.A B A B
8. A fair coin is tossed 3n times. Find the probability that the number of heads is twice the number of tails. Expand your answer using Stirling’s formula.
7. A single card is removed at random from a deck of 52 cards. From the remainder we draw two cards at random and find that they are both spades. What is the probability that the first card removed
6. Urn I contains 4 white and 3 black balls, and Urn II contains 3 white and 7 black balls. An urn is selected at random, and a ball is picked from it. What is the probability that this ball is
5. Two fair dice are thrown. Let A be the event that the first shows an odd number, B be the event that the second shows an even number, and C be the event that either both are odd or both are even.
This is sometimes called Bonferroni’s inequality, but the term is not recommended since it has multiple uses.
1. A fair die is thrown n times. Show that the probability that there are an even number of sixes is 1 21 + ( 2 3 )n. For the purpose of this question, 0 is an even number.2. Does there exist an
The following comparison of surgical procedures is taken from Charig et al. (1986). Two treatments are considered for kidney stones, namely open surgery (abbreviated to OS) and percutaneous
A coin is tossed 2n times. What is the probability of exactly n heads? How does your answer behave for large n?Solution The sample space is the set of possible outcomes. It has 22n elements, each of
(e) If you choose first, what is the probability that you survive, given that your sister survives
(d) Is it in your best interests to persuade your sister to choose first?
(c) If you choose first and die, what is the probability that your sister survives?
(b) If you choose first and survive, what is the probability that your sister survives?
(a) If you choose before your sister, what is the probability that you will survive?
You are travelling on a train with your sister. Neither of you has a valid ticket, and the inspector has caught you both. He is authorized to administer a special punishment for this offence.He holds
A biased coin shows heads with probability p = 1 − q whenever it is tossed. Let un be the probability that, in n tosses, no two heads occur successively. Show that, for n ≥ 1, un+2 = qun+1 + pqun
2 , and you pick a ball at random from the chosen urn. Given the ball is black, what is the probability you picked Urn I?
Here are two routine problems about balls in urns. You are presented with two urns. Urn I contains 3 white and 4 black balls, and Urn II contains 2 white and 6 black balls.(a) You pick a ball
Exercise 1.36 Consider the experiment of tossing a fair coin 7 times. Find the probability of getting a prime number of heads given that heads occurs on at least 6 of the tosses.
Exercise 1.35 Show that P(B | A) = P(A | B)P(B)P(A)if P(A) > 0 and P(B) > 0.
Exercise 1.34 If (,F , P) is a probability space and A, B,C are events, show that P(A ∩ B ∩ C) = P(A | B ∩ C)P(B | C)P(C)so long as P(B ∩ C) > 0.
Exercise 1.30 Which of the following is more probable:(a) getting at least one six with 4 throws of a die,(b) getting at least one double six with 24 throws of two dice
Exercise 1.29 Show that the probability that a hand in bridge contains 6 spades, 3 hearts, 2 diamonds and 2 clubs is 13 613 313 2252 13.
Exercise 1.28 Show that the probability that two given hands in bridge contain k aces between them is4 k48 26 − k52 26.
Exercise 1.27 In a game of bridge, the 52 cards of a conventional pack are distributed at random between the four players in such a way that each player receives 13 cards. Show that the probability
(a) there are nr possible arrangements,(b) there are????r k(n − 1)r−k arrangements in which the first cell contains exactly k balls
Exercise 1.26 We distribute r distinguishable balls into n cells at random, multiple occupancy being permitted. Show that
sequences of possible outcomes in which exactly k heads are obtained. If the coin is fair (so heads and tails are equally likely on each toss), show that the probability of getting at least k heads
We usually write P(ω) for the probability P({ω}) of an event containing only one point in .Example 1.23 (Equiprobable outcomes) If = {ω1,ω2, . . . ,ωN } and P(ωi ) = P(ωj ) for all i and j
in whether or not this given ω is the actual outcome of E; then we require that each singleton set {ω} belongs to F . Let A ⊆ . Then A is countable (since is countable), and so A may be
1.5 Discrete sample spaces Let E be an experiment with probability space (,F , P). The structure of this space depends greatly on whether is a countable set (that is, a finite or countably
In the first case your event space should have 2210 events, but in the second case it need have only 211 events.
Describe the appropriate probability space in detail for the two cases when(a) the outcome of every toss is of interest,(b) only the total number of tails is of interest.
Exercise 1.22 A fair coin is tossed 10 times (so that heads appears with probability 1 2 at each toss).
By drawing a Venn diagram or otherwise, find the probability that exactly two of the events A, B, C occur.
P(A ∩ B) = 3 10 , P(B ∩ C) = 4 10 , P(A ∩ C) = 2 10 , P(A ∩ B ∩ C) = 1 10 .
Exercise 1.21 Let A, B,C be three events such that P(A) = 5 10 , P(B) = 7 10 , P(C) = 6 10 ,
Exercise 1.20 If A, B,C ∈ F , show that P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(A ∩ C) + P(A ∩ B ∩ C).
Exercise 1.19 If A, B ∈ F , show that P(A \ B) = P(A) − P(A ∩ B).
(c) The third condition (1.4) is written in terms of unions. An event space is also closed under the operations of taking finite or countable intersections. This follows by the elementary formula (A
(a) An event space F must contain the empty set ∅ and the whole set . This holds as follows. By (1.2), there exists some A ∈ F . By (1.3), Ac ∈ F .We set A1 = A, Ai = Ac for i ≥ 2 in (1.4),
1.15. In the case of (fl, 9'3, 111), suppose that J) 61 and the nth partition is obtained by dividing '1/ into 2n equal parts: what are the XII'S in (21)? Use this to "explain away" the St. Peterburg
1.*14. Prove that if J) is a measure on ::--'ftoo that is singular with respect to?J?, then the Xn 's in (21) converge a.e. to zero. [HINT: ShO'.v thatand apply Fatou's lemma. {Xn} is a
1.13. Prove the inequality (19).
1.12. In Example (VI) show that (i) f} is generated by 1]; (ii) the events{En, n EN} are conditionally independent given 1]; (iii) for any l events Enj , 1 :::. j :::. l and any k :::. l we have
1.11. Deduce from Exercise 10 that g(suPn ISn lin) < 00 if and only if #(IX11Iog+IXI!)
1.10. Let {Xn, n E N} be a sequence of independent, identically distributed r.v.'s with {(IXI /) < 00; and Sn = 2:j=1 Xj . Define a = inf{n ~ 1: IXnl >n}. Prove that if J((ISal/a)l(a
1.9. Let {Y k' 1 S k S n} be independent r. v. 's with mean zero and finite. 2 varIances CJ k;k k Sk = LYj, S~ = LCJ;>O, Zk = S~ -s~j=l j=l Prove that {Zk. 1 S k S n} is a martingale. Suppose now all
1.*8. As an example of a more sophisticated application of the martingale convergence theorem, consider the following result due to Paul Levy. Let{Xn, 11 E NO} be a sequence of unifonnly bounded
1.*7. The analogue of Theorem 9.5.4 for p = 1 is as follows: if Xn ~ 0 for all n, then e /{supXn} S --[1 + supJ'{Xn10g+Xn}], n e -1 n where log + x = log x if x ~ 1 and 0 if x S l.
1.6. Let f be a real bounded continuous function on :li1 and J-t a p.m. on}')l1 such that V'x E gel: f(x) = r f(x + y)J-t(dy). )9{1 Then f (x + s) = f (x) for each s in the support of J-t. In
1.*5. In the notation of Theorem 9.6.2, suppose that there exists 0>0 such that then we have 2P{X j E A j 1.0. and X j E B j 1.o.} -U.
1.4. Let {Xn, n EN} be an arbitrary stochastic process and let~' be as in Sec. 8.1. Prove that the remote field IS almost trIvIal If and only If for each A E ::000 we have lim sup IP(AM) -P(A)9(M)1 =
1.3. Let {Zn, 11 E NO} be positive integer-valued r.v.'s such that Zo = 1 and for each n ~ 1, the conditional distribution of Zn given Zo, ... ,Zn-1 is that of Zn-1 independent r.v.'s with the common
1.2. Suppose that for each n, {X j, 1 :s j < n} and {Xi, 1 :s j :s n} have respectively the n-dimensional probability density functions Pn and qn' Define qn(X1, ••• ,Xn) yn=------Pn(X1,.·.,
1.*1. Suppose that {Xn, n EN} is a sequence of integer-valued r.v.'s having the following property. For each n, there exists a function Pn of n integers such that for every k € .V, ',lie haveif the
1.*14. In Exercise 13 let Q be the set of rational numbers in [0, 1]. For each t E (0, 1) both limits below exist a.e.:limXs ,.Itt I(:Q[HINT: Let {Qn, n .:::: I} be finite subsets of Q such that Qn t
1.13. Let {X" ~7;; t E [0, I]} be a continuous parameter supennartingale.For each t E [0, 1] and sequence {t11} decreasing to t, {X,,,} converges a.e. and in Ll. For each t E [0, 1] and sequence
1.12. Let {~n, n E N} be a sequence of independent and identically distributed r.v.'s with zero mean and unit variance; and Sn = '2:'j=1 ~j.Then for any optional LV. a relative to {~n} such that
1.*11. If {Xn} is a martingale or positive submartingale such that sUPn "(X~) < 00, then {Xn} converges in L 2 as "vell as a.e.
1.10. Let {Xn, ;:"/n} be a potential; namely a positive supennartingale such that limn~oo {(X,I) = 0; and let Xn = Y,z -Zn be the Doob decomposition[cf. (6) of Sec. 9.3]. Show that
1.9. Let {Xn,,/fr/1;n EN} be a supennartingale satisfying the condition limn-->x /' (XI1 ) > -00. Then we have the representation XI1 = X;I + X;; where{X;I' ;'frl1 }is a martingale and {X;;, /In} is
1.8. Let {XIl , .~, n EN} be a submartingale and let ex be a finite optional r.v. satisfying the conditions: (a) / (IXal) < 00, and (b)lim r IXI1 I d./j) = O. '1~00 J{a>n}Then {X aM, '~/\Il; 11 E N
1.7. Pwve a result fOI cIosabiIity [on the left] which is similar to Exercise 6 but for the index set -N. Give an example to show that in case of N we may have limll~ex; I(XIl ) =1= / (Xoo), whereas
1.*6. A smartingale {Xn, ~In; n EN} is said to be closable [on the right]iff there exists a r v Xoe such that {XI1' Jin; n E Noe} is a smartingale of the same kind. Prove that if so then we can
1.*5. Every L I-bounded martmgale is the difference of two positive L 1_ bounded martingales This is due to Krickeberg [HINT' Take one of them to
1.*4. As a sharpening of Theorems 9.4.2 and 9.4.3 we have, for a positive supermartingale {XIl' ,~IIl' n EN}:,J7J{v(n) > k} < cf(Xl 1\a) (~)k-l[a.b] --b b ' -{-(Il) k} r(X l 1\ b)(a)k-l -f>v > < -,
1.3. Generalize the upcrossing inequality for a submartingale {Xn, 2l,;} as follows:Similarly. generalize the downcrossing inequality for a positive supermartin gale {Xn, (-(n)-Xl 1\ b cr {v[a b] I
1.2. Let {X n} be a positive supermartingale. Then for almost every w, Xk(W) = 0 implies Xn (w) = 0 for all n::: k. [This is the analogue of a minimum principle in potential theory.]
1.*1. Prove that for any smartingale, we have for each ),>0:For a martingale or a positive or negative smartingale the constant 3 may be replaced by 1. P{sup X}
1.17. As in the preceding exercise, deduce a new proof of Theorem 9.3.4 by taking Vn -1 {a
1.*16. Let {Xn,S?;;} be a martingale: Xl = Xl, Xn = Xn -Xn-l for n :::. 2;let Vn E 3?;;-1 for n :::. 1 where.% = .9l; now put j=lShow that {Tn, g;;} is a martingale provided that Tn is integrable for
1.15. Prove that for any L I-bounded smartingale {Xn, g;;;, n E N oo}, and any optional ex, we have 6(IXa I) < 00. [HINT: Prove the result first for a martingale, then use Doob's decomposition.]
1.14. In the gambler's ruin problem, suppose that S I has the distribution POI + (1 -P)O-l' P =1= ~;and let d = 2p -1. Show that 0'(Sy) = dg(y). Compute the probabilities of ruin by using difference
1.* 13. In the gambler's ruin problem take b = 1 in (20). Compute i (S {J/\Jl )for a fixed n and show that {So, S {J/} forms a martingale. Observe that {So, S {J}does not form a martingale and
1.12. Apply Exercise 11 to the gambler's ruin problem discussed in the text and conclude that for the a in (20) we must have J'(a) = +00. Verify this by elementary computation.
1.*11. Let {Xn, 9?;;; n E N} be a [super]martingale satisfying the following condition. there exists a constant lv! such that for every n ::: 1:0"{lXn - Xn-l ll:16n-l } ::: Ma.e.where Xo = 0 and .%
1.10. If {Xn} is a uniformly integrable submartingale, then for any optional r.v. a we have(i) {Xai} is a uniformly integrable submartingale;(ii) 0'(X 1) ::: 0'(Xa) ::: SUpn 0'(Xn).
1.*9. Prove that if {Y n, ,'j6n} is a martingale such that Y n E ,'j6n-l, then for every n, Y n = Y 1 a.e. Deduce from this result that Doob's decomposition (6)is unique (up to equivalent r.v.'s)
1.8. Find an example of a martingale {Xn} such that Xn -+ -00 a.e. This implies that even in a "fair" game one player may be bound to lose an arbitrarily large amount if he plays long enough (and no
1.7. Find an example of a positive martingale which is not uniformly inte grable. [HINT: You win 2n if it's heads n times in a row, and you lose everything as soon as it's tails.]
1.6. If {X/1, .-0/1} and {X~, .-'l'n} are martingales, then so is {Xn + X~, ~}.But it may happen that {X /1} and {X~} are martingales while {X n + X~} is not.[HINT: Let XI and xi be independent
1.S. Every sequence of integrable LV.'S is the sum of a supermartingale and a submartingale.
1.4. Suppose each Xn is integrable and J{Xn+1 I XI," .,Xn} = n-I(XI + ... +Xn)then {(n -I )(X I+ ... + Xn), n E N} is a martingale.
1.3. S oppose {X~l'), '~l}, k -1, 2, are two [soperJmartingales, ex is a finite that {X/1' .:1'n} is a [super]martingale. [HINT: Verify the defining relation in (4)for m 11+ 1.]
1.*2. If X is an integrable r.v., then the collection of (equivalence classes of) LV.'S l(X I §) with r; ranging over all Borel subfields of g;, is uniformly integrable.
1.1. The defining relation for a martingale may be generalized as follows.For each optional LV. ex < 11, we have ${Xn I ~} = Xa' Similarly for a smartingale.
1.16. The same conclusion is true if the random walk above is replaced by a homogeneous Markov process for which, e.g., there exist 0>0 and 17>0 such that P(x, a l (x 8, x + 8)) ~ 'I for every x.
1.15. If {S n, n E N} IS a random walk such that .o/'{S 1 #= O} > 0, then for any finite interval [a, b] there exists an E < 1 such that::IP{Sj E [a, b], 1 < j < n} < Ell This is just Exercise 6 of
1.14. Let {Xn, n E NO} be an independent process. Lets(r+l)n j=O for r > 1. Then {S~), n E NO} has the rth-order Markov property. For r = 2, give an example to show that it need not be a Markov

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