INDIAN STATISTICAL INSTITUTE

INDIAN STATISTICAL INSTITUTE B1/BVR Probability Home Assignment 1 20-07-07 1. A poker hand means a set of five cards selected at random from usual deck of playing cards. (a) Find the probability that it is a Royal Flush - means that it consists of ten, jack, queen, king, ace of one suit. (b) Find the probability that it is four of a kind - means that there are four cards of equal face value. (c) Find the probability that it is a full house - means that it consists of one pair and one triple of cards with equal face values. (d) Find the probability that it is a straight - means that it consists of five cards in a sequence regardless of suit. (e) Find the probability that it consists of three cards of equal face value and two other cards but not a full house. (f) Find the probability that it consists of two distinct pairs and another card but does not fall into previous categories. (g) Find the probability that it consists a pair and three other cards but does not fall into previous categories. 2. A bridge distribution means a distribution of the usual deck of playing cards among four persons to be called N, E, S, W, each getting 13 cards. (a) Show that the probability p of W receiving exactly k aces is same as the probability that an arbitrary hand of 13 cards contains excatly k aces. (b) What is the probability that N and S together get k aces? Here k =0,1,2,3,4. (c) Find the probability that N,S,E,W get a, b, c, d spades respectively. 3. I have n sticks. Each is broken into two pieces - one long and one short piece. These 2n pieces are paired at random two form n sticks. (a) What is the probability that they are joined to form original sticks? (b) Find the probability that all long parts are paired with short parts. 4. In how many ways can two rooks of different colours be put on a chess board so that they can take each other? 5. show that it is more probable to get at least one ace with four dice than at least one double ace in 24 throws of two dice. This is apparently known as de Mere s paradox. There is, of course, no paradox it just 1

so happens that Chevalier de Mere thought that both probabilities are equal. 6. If n balls are placed at random into n cells, find the probability that exactly one cell remains empty. 7. A man is given n keys, in a random order, of which only one fits the door. He tries the keys, one after the other, to open the door. This procedure may require 1,2,3, n trials. Show that each of these n has probability 1/n. 8. A box contains 90 good and 10 defective items. If 10 items are selected then what is the probability that none of them is defective? 9. If n men among whom are A and B, stand in a row what is the probability that there are exactly r men between A and B? What if they stand in a ring (what does this mean?) and the clock-wise direction is used for counting the number between A and B? 10. What is the probability that two throws with three dice each will show the same configuration, if the dice are distinguishable? What if the dice are not distinguishable? 11. From a population of n elements a sample of size r is taken. Find the probability that none of N prescribed elements are included in the sample. Assume that the sampling is with replacement. What if it is without replacement? 12. What is the probability that the birthdays of 12 people will fall in 12 different calender months assume equal probabilities for the 12 months. What is probability that the birthdays of 6 people will fall in exactly two calender months. What is the probability that the birthdays of 30 people fall as follows: six months contain two birthdays each and six months contain three birthdays each. 13. A closet contains n pairs of shoes. 2r shoes are chosen at random. What is the probability that there will be no complete pair? exactly one complete pair? exactly two cpmplete pairs? 2

14. A group of 2N boys and 2N girls is divided into two equal groups. Find the probability that each group will have equal number of boys and girls. 15. There is a three volume dictionary among 40 books arranged in a shelf in a row. Find the probability that the volumes are arranged in the increasing order from left to right ( not necessarily side by side ). 16. A natural number X is chosen at random from {1, 2,, n}. Find the probability p n that X is divisible by k. natural number. Find the limit of p n as n. Here k is a fixed Find the probability that when divided by k, the selected number X leaves a remainder r. Evaluate the limit of this probability as n. Suppose a 1 and a 2 are fixed coprime natural numbers. Find the probability p n that the selected number is not divisible by a 1 and not divisible by a 2. Find the limit of p n as n. What if you had more than two pair-wise coprime numbers and ask the same question. Calculate the probability p n that X 2 1 is divisible by 10. Evaluate limit of p n. 17. A number X is chosen at random from the set {0, 1, 2,, 10 n 1}. Find the probability that X is a k-digit number. A number a is a k- digit number if it is of the form a = k 1 0 a i 10 i where a i are integers from 0 to 9 and a k 1 is not zero. 18. Two numbers X and Y are chosen at random with replacement from {1, 2,, n}. Which of the following events is more probable : [X 2 Y 2 is divisible by 2] OR [X 2 Y 2 is divisible by 3]? 19. Two sets A 1 and A 2 are chosen with replacement from the collection of all subsets of the set {0, 1,, n}. Find the probability that A 1 and A 2 are disjoint. Suppose that I select r subsets with replacement. What is the probability that they are pairwise disjoint. 20. Ten numbers are selected at random without replacement from among {1, 2,, 30}. Find probabilities of the following events: A : All numbers are odd. B : Exactly five numbers are divisible by 3. 3

C : 5 numbers are odd and 5 are even and exactly one divisible by 10. 21. Two numbers are selected at random from {1, 2,, N} without replacement. Find the probability that the first number is larger than the second. Suppose that three numbers are selected without replacement from the same set as above. Find the probability that the third number is between the other two. Find the probability that the second number is between the other two. From the same set as above n numbers are selected without replacement. Suppose that X is the largest of the numbers selected and Y is the second largest. Find the probability that Y M < X. Here M is a given integer. Suppose that M and N increase to infinity in such a way that M/N α where α is a number between 0 and 1, Find the limit of your probability. What if X is the (m + 1) th and Y is m th when the selected numbers are arranged in increasing order. 22. One mapping, from the set of all mappings of {1, 2,, n}, is selected at random. What is the probability that the selected mapping transforms each of the n elements into 1? What is the probability that element i has exactly k preimages? ( Here i and k are preassigned ). What is the probability that the element i is transformed to j? What is the probability that the elements i 1, i 2 and i 3 (assume distinct) are transformed to j 1, j 2 and j 3 respectively? What is the probability that the selected mapping is one-to-one? What is the probability that the selected mapping is onto? What is the probability that the range of the map consists of exactly k points? 23. One permutation is selected at random from the set of all permutations of {1, 2,, n}. What is the probability that the identity permutation is chosen? What is the probability that the selected permutation transforms i 1, i 2,i k to j 1, j 2,,j k respectively? What is the probability that the permutation keeps i fixed? What is the probability that the elements 1, 2, and 3 form a cycle in that order? in some order? What is the probability that all the elements form a cycle? What is the probability that there are exactly two cycles? 4

24. What is the probability of getting a total of 20 points with 6 dice? 25. A box contains a white and b black balls. Balls are drawn one by one without replacement until balls of the same colour are left in the box. What is the probability that the balls left are all white? 26. A box contains N tickets numbered 1, 2,, N. From this box n tickets are drawn. What is the probability that the largest number drawn is M? These problems are from the books of Feller, Sevastyanov et al and Uspensky. You should first see the book of Feller and try all the problems. Hoel, Port and Stone (three authors) is another book. You should go to the library and just have a look at probability books. For a better understanding, you should all discuss among yourselves. 5

B1/BVR Probability Home Assignment 1 30-07-07 27. Given P (A) = 1/3, P (B) = 1/4, and P (A B) = 1/6, find the following probabilities: P (A c ); P (A c B), P (A B c ); P (A B c ); P (A c B c ). 28. Given P (A) = 3/4, P (B) = 3/8, show that 1/8 P (A B) 3/8. 29. If P (A) = 0.9 and P (B) = 0.8 show that P (A B) 0.7. In general show that, P (A B) P (A) + P (B) 1 for any two events. This is known as Bonferroni inequality. (Carlo Emilio Bonferroni was an Italian Mathematician/Statistician; 1892-1960). 30. For any n events, A 1,, A n, show that P ( A i ) P (A i ). This is known as Boole s inequality. (George Boole was a British Mathematician/Philosopher; 1815-1864). 31. For any n events, A 1,, A n, show that P ( A i ) = S 1 S 2 + S 3 ± S n. where S 1 = P (A i ), S 2 = i<j P (A i A j ) etc. We proved this in case all the outcomes are equally likely. This is true even when the outcomes are not equally likely. This is known as Poincare s theorem or the inclusion-exclusion principle. (Jules Henri Poincare was a French Mathematician/Mathemetical Physicist/Philosopher; one of the greatest Mathematicians; 1854-1912). 32. Three winning tickets are drawn from an urn containing 100 tickets. What is the probability of winning for a person who bought 4 tickets? What is the probability if he bought only one ticket? 33. A bakery makes 80 loaves of bread daily of which ten are underweight. An inspector weighs five loaves at random. What is the probability that an underweight loaf will be discovered? 34. The coefficients of the quadratic equation ax 2 + bx + c = 0 are determined by throwing a die three times. Find the probability that the roots are real. Find the probbility that the roots are complex. 6

35. An urn contains nr balls numbered 1, 2,, n where each number i, 1 i n, appears on r balls. From the urn, N balls are drawn without replacement. Find the probability that exactly m numbers will appear in the sample. Find the probability that each of the n numbers appears at least once in the sample. 36. There is no probability here. Write proofs of the following statements: (A B) c = A c B c ; and (A B) c = A c B c These are called De Morgan s Laws. (Augustus De Morgan was Indian born British Logician, 1806-1871). What if you have a finite number of sets. many(?) sets. What if you have many 37. Workout all problems from sections 10 and 11, chapter 2 of Feller. (William Feller was Croatian/Austrian born American Probabilist; 1906-1970). Some problems in this set are from the book of T. Cacoullos. You should not only workout the problems but also write down your solutions. This will help you learn how to communicate your argument. 7

38. At a parking lot there are 12 places arranged in a row. A man observed that there are 8 cars parked and that the four empty places are adjacent to each other. Given that there are four empty places, is this arrangement surprising? 39. Three dice are rolled. Given that no two show the same face, find the conditional probability that at least one is an ace. 40. Ten dice are thrown once. Given that there is at least one ace, find the conditional probability that there are two or more aces. 41. In Bridge West has no ace. What probability should be attributed to the event of his partner having no ace? partner having two or more aces? 42. In a bolt factory, machines A, B and C manufacture 25, 35, and 40 percent of the total. Of their outputs 5, 4, and 2 percent are defective. A bolt is selected at random and is found to be defective. Given this what is the probability it was manufactured by machine A? by B? by C? 43. In a population where half are males and half are females, it is known that 5 men out of 100 and 25 women out of 10,000 are color blind. If a person is selected at random and is found to be color blind, what is the conditional probability that the person is a male. 44. A die is thrown as long as necessary for an ace to turn up. Assuming that the ace does not turn up at the first throw, what is the (conditional) probability that more than three throws will be necessary? 45. A die has four red faces and two white faces and a second die has two red and four white faces. A fair coin is flipped once. If it falls heads we keep on throwing the first die and if it falls tails we keep on throwing the second die. Show that the probability of red at any throw is 1/2. If the first two throws resulted in red what is the conditional probability of red at the third throw. If red turns up at the first n throws, what is the conditional probability that first die is being used? 8

B1/BVR Probability Home Assignment 4 08-9- 2000 46. Let the events A 1, A 2, A n be independent and P (A k ) = p k. Find the probability p that none of the events occurs. Show that always p < exp( p k ). 47. Two numbers are selected at random with replacement from {0, 1, 2, 9}. Their sum denoted in decimal notation as 10 X + Y. Find the joint distribution of X and Y. Are they independent? What if instead of sum, we calculated product? 48. Have a coin whose chance of heads is p in a single toss. I keep on tossing the coin until k heads appear. Let X 1 be the number of tails before the first head, X 2 be the number tails between the first and second head etc, X k be the number of tails between the k 1 th and k th head. Find their joint distribution. Are they independent? 49. I keep on rolling a fair die independently until each face appears at least once. Let X 1 be the first number that appears and Y 1 = 1. Let X 2 be the second number that appears which is different from X 1 and Y 2 be the number of the throw. Similarly define X i and Y i for i = 3, 4, 5, 6. Find the joint distribution of (X 1,, X 6 ). Define the waiting times W 1 = X 1 and for 2 i 6, W i = X i X i 1. Find the joint distribution of (W 2, W 6 ). 50. A box has M white balls and N M black balls. Balls are drawn without replacement from the box. Let X 0 be the number of black balls before the appearance of a white ball; X 1 be the number of black balls between the first and second white balls etc and finally X M be the number of black balls after the last white ball. Find the distribution of X 0. Find the joint distribution of X 0 and X 1. Find the joint distribution of X 0, X 1 and X 2. Find the expected value of X i. 51. We roll a die until we obtain a total of at least five points. Let X be the points on the last throw and Y be the number of throws needed. Find their joint distribution. Are they independent? 52. X and Y are independent random variables each taking finitely many integer values. Let Z = X + Y. Let a be the least value ( with positive 9

probability) of Z and b be the largest value ( with positive probability) of Z. Show that either P (Z = a) 1/4, or P (Z = b) 1/4. 53. A random variable X assumes only non-negative integer values. Show that E(X) = P (X k) 54. An urn contains M 1 balls numbered 1, M 2 balls numbered 2,, M N balls numbered N. We draw n balls without replacement from the urn. Let S denote the sum of all the numbers from among {1, 2,, N} that did not appear in the sample. Find its mean. 55. There are 25 students in a group. Assume that birthday could fall, independent of others, in any of the 12 months at random. Let X be the number of months in which there is no birthday. Find its mean and variance. 56. There are n particles each thrown, independent of others, into one of N boxes at random. Let X denote the number of empty boxes. Find its mean µ and variance σ 2. Calculate the limit of µ/n and σ 2 /N as n and N tend to infinity in such a way that n/n α. 57. We have a large supply of balls and N boxes. We through each of the balls, independent of others into the boxes at random. Fix an integer k N. Stop throwing when you see exactly k boxes occupied. Let X be the number of balls thrown. Find its expectation and variance. 58. Three numbers are selected at random without replacement from the set {1, 2,, N}. Find the conditional probability that the third number falls between the first two numbers given that the first nymber is smaller than the second. 59. two fair dice are thrown. Find the conditional probability that two fives occured given that the total is divisible by five. 60. One card is selected at random from 100 cards numbered 00,01,,98,99. Let X and Y be the sum and product respectively of the digits on the selected card. Find their joint distribution and all conditional distributions. k=1 10

61. Two subsets A 1 and A 2 are selected at random with replacement from the collection of all subsets of {1, 2,, N}. Find the conditional probability that A 1 has k elements and A 2 has l elements given that A 1 and A 2 are disjoint. 11

B1/BVR Probability Home Assignment 5 24-10 - 2000 62. The random variables X and Y are independent G(p). That is for k 0, P (X = k) = p q k. Find the following. P (X = Y ) P (X > Y ) P (X < Y ) P (X = k X > Y ) P (X = k X < Y ) P (X = k X = Y ) P (X = k X + Y = l) E(X X + Y = l). 63. The random variables X 1, X 2,..., X N are independent Poisson with parameters λ 1, λ 2,..., λ N respectively. Find the conditional distribution and conditional expectation of X 1 +X 2 +...+X k given X 1 +X 2 +... + X N = l. Here k < N. 64. From an urn containing M white and N M black balls we draw without replacement a sample of size n. From this sample we draw a sample of size m without replacement with m < n. Find the distribution of the number of white balls in the second sample. Instead suppose we draw a sample of size m from the urn without replacement. Find the dstribution of the number of white balls in the sample. 65. I have a coin whose cahnce of heads in a single toss is p. I keep on tossing it until I get a two successive heads and then stop. Find the distribution of the number of tosses needed. 66. An urn contains 3 white, 5 black and 2 red balls. Two persons draw balls in turn, without replacement. The first person to draw a white ball before the appearance of a red ball wins the game. However, if a red ball is drawn by anyone before the appearance of a white ball, then the game is declared a tie. Calculate the probability that the person who begins the game wins. Calculate the probability that the other person wins the game. Calculate the probability that the game ends in a tie. 67. Consider the polya Urn scheme with b black balls and r red balls and after each draw, c balls of that color are added. Given that the second ball was black, what is the probability that the first ball was black? 12

Write a complete proof of the fact that probability of a black ball at any draw is b/(b + r). In what follows m < n are positive integers. Prove that the probability that m th and n th draws produce black, black is. Prove that the probability that m th and n th b(b+c) (b+r)(b+r+c) draws produce black, red is br. (b+r)(b+r+c) Prove that in the first m + n drawings we get m black and n red balls is same as the probability that the first m draws are red and the next n draws are black. Let A and B each stand for either red or black. Show that the probability of A at the n th draw given B at the m th draw is same as the probability of A at the m th draw given B at the n th draw. Suppose that p k (n) denotes the probability of k black balls in the first n draws. Prove the recurrence relation : p k (n + 1) = p k (n) r + (n k)c b + r + nc + p k 1 (n) Here p 1 (n) is to be interepreted as zero. Deduce that Set so that b b + r = p, p k (n) = ( b c ) k ( r c ) n k ( b+r c r b + r = q, ) n c b + r = γ p k (n) = ( p γ ) k ( q γ ) n k ( 1 γ ) n b + (k 1)c b + r + nc As long as p > 0, q > o, γ > 1, the above formula gives a legitimate probability distribution on {0, 1,, n} for each fixed n. This is called Polya distribution ( with those parameters). If n, p 0, γ 0 in such a way that np λ and nγ 1/ρ, then show that ( ) λρ + k 1 ρ 1 p k (n) ( k 1 + ρ )λρ ( 1 + ρ )k for each k 0. This is called negative binomial distribution. 13