14/15

P. 1

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE

STAT1801/2901 Probability and Statistics: Foundations of Actuarial Science

Assignment 1

Due Date: February 27, 2015 (Hand in your solutions for Questions 9, 16, 19, 28, 31, 39, 43, 46, 50, 56) 1. How many (possibly overlapping) squares are in an nm grid? Assume that nm and all

the squares have their edges parallel to the edges of the grid. 2. A total of 7 different gifts are to be distributed among 13 children. How many distinct results

are possible if no child is to receive more than one gift? 3. How many n-bit binary strings (i.e. strings consists of 0 and 1 only) have exactly k transitions

(a transition is an adjacent pair of dissimilar bits, i.e. 01 or 10)? 4. From a group of 8 women and 6 men a committee consisting of 4 men and 4 women is to be

formed. How many different committees are possible if (a) 2 of the men refuse to serve together; (b) 2 of the women refuse to serve together; (c) 1 man and 1 woman refuse to serve together?

5. In how many ways can 12 people be seated in a row if

(a) there are no restrictions on the seating arrangement; (b) persons A and B must sit next to each other; (c) there are 6 men and 6 women and no 2 men or 2 women can sit next to each other; (d) there are 5 men and they must sit next to each other; (e) there are 6 married couples and each couple must sit together?

6. Four separate awards (best scholarship, best leadership qualities, and so on) are to be

presented to selected students from a class of 25. How many different outcomes are possible if (a) a student can receive any number of awards; (b) each student can receive at most 1 award?

7. You are in a game of Russian roulette with an opponent, but this time the gun (a 6 shooter

revolver) has three bullets in three consecutive chambers. The barrel is spun only once. You and your opponent would then take turns to point the gun to your own head and pulls the trigger. The game stops when any one of you dies. Would you prefer to be the first or second to shoot?

8. From a group of 4 lawyers, 8 accountants, and 7 doctors, a committee of three is selected at

random. What is the probability that the committee has more lawyers than doctors? 9. If 8 castles are randomly placed on a chessboard, compute the probability that none of the

castles can capture any of the others. That is, compute the probability that no row or column contains more than one castle.

14/15

P. 2

10. (Poker)

If it is assumed that all

5

52 poker hands are equally likely, what is the probability of being

dealt

(a) one pair? (This occurs when the cards have denominations a, a, b, c, d where a, b, c, and d are all distinct.)

(b) two pairs? (This occurs when the cards have denominations a, a, b, b, c, where a, b, and c are all distinct.)

(c) three of a kind? (This occurs when the cards have denominations a, a, a, b, c.) (d) a straight? (Five consecutive cards of any suits which are not all the same, including 10, J,

Q, K, A.) (e) a flush? (Five cards of the same suit and not all in consecutive order.) (f) full house? (This occurs when the cards have denominations a, a, a, b, b.) (g) four of a kind? (This occurs when the cards have denominations a, a, a, a, b.) (h) a straight-flush? (Five consecutive cards of the same suit, including 10, J, Q, K, A.)

11. Six couples are arranged to sit at a round table. How many arrangements can be made so that

(a) a particular couple is together? (b) all couples are together? (c) no two ladies may be together?

12. Prove that

r

n

r

n

r

n 1

1

(This identity is the basis of a special arrangement of numbers in a triangular array known as the Pascal Triangle. Using this identity together with the mathematical induction, one can easily prove that all the binomial coefficients must be integers.)

13. Prove that

0110

m

r

n

r

mn

r

mn

r

mn .

Hint: Consider a group of n men and m women. How many groups of size r are possible?

14. Use exercise 13 to prove that

n

k k

n

n

n

0

22

.

15. Show that for 0n ,

010

n

i

i

i

n.

Hint: Use the binomial theorem.

16. How many possible unordered outcomes are there when 5 six-faced dice are rolled?

14/15

P. 3

17. We have 25 thousand dollars that must be invested among 4 possible opportunities. Each investment must be integral in units of 1 thousand dollars, and there are minimal investments that need to be made if one is to invest in these opportunities. The minimal investments are 3, 4, 4, and 5 thousand dollars. How many different investment strategies are available if (a) an investment must be made in each opportunity; (b) investments must be made in at least 3 of the 4 opportunities?

18. In how many ways can we fill a bag with n fruits subjects to the following constraints:

the number of apples must be even; the number of bananas must be a multiple of 5; there can be at most four oranges; there can be at most one pear.

19. Six fair dice each numbered {1, 2, 2, 3, 3, 3} are rolled simultaneously. What is the

probability that the sum will be equal to 10? (Hint: Use the generating function 32 32 xxx .)

20. Two six-face dice are thrown. Let E be the event that the sum of the dice is odd; let F be the

event that at least one of the dice lands on 1; and let G be the event that the sum is 5. Describe the events FE , FE , GF , CFE , and GFE .

21. Let E, F, G be three events. Express the following events by set notations.

(a) only E occurs; (b) both E and G but not F occurs; (c) at least one of the events occurs; (d) at least two of the events occurs; (e) all three occur; (f) none of the events occurs; (g) at most one of them occurs; (h) at most two of them occur; (i) exactly two of them occur; (j) at most three of them occur.

22. Suppose that A and B are mutually exclusive events for which 4.0AP and 5.0BP .

What is the probability that (a) either A or B occurs; (b) A occurs but B does not; (c) both A and B occur?

23. In the following figure of pipes, assume that the probability of each valve being opened is p

and that the statuses of all the valves are mutually independent. Find the probability that water can flow from L to R.

L R

14/15

P. 4

24. Consider the following experiment:

“Two fair coins are tossed and the outcome is observed.”

Hector is trying to formulate this experiment using the language of probability but is having some difficulty. He chooses the sample space to be

TTHTTHHH ,,,,,,,

and the event space to be the collection of subsets of . Hector observes that the assumption that the first coin is fair is equivalent to the statement

5.0,,, THHHP ,

and likewise, the assumption that the second coin is fair is equivalent to the statement

5.0,,, HTHHP .

He convinces himself that this assignment of probability should be sufficient to completely define a probability measure but has difficulty calculating EP for some of the events E . (a) Using appropriate properties of probability measures, show that

5.0,,, TTHTP , 5.0,,, TTTHP

and interpret these statements.

(b) Show that Hector’s formulation does not uniquely determine the probability measure. (c) Under Hector’s formulation, what are the possible values of HHP , ? Does it

surprise you that HHP , need not be 0.25? (d) What condition has Hector neglected to include in his formulation? What additional

condition on the probability measure is sufficient to obtain the correct model? (e) Can you think of a different experiment whose formulation satisfies all of Hector’s

conditions but for which 25.0, HHP ? 25. A certain town of population size 200,000 has 3 newspapers : I, II, and III. The proportion of

townspeople that read these papers are as follows: I: 15% I and II: 7% I and II and III: 1% II: 28% I and III: 3% III: 4% II and III: 2%

(The list tells us, for instance, that 14000 people read newspapers I and II.)

(a) Find the number of people reading only one newspaper. (b) How many people read at least two newspapers? (c) If I and III are morning papers and II is an evening paper, how many people read at least

one morning paper plus and evening paper? (d) How many people read only one morning paper and one evening paper?

14/15

P. 5

26. The following data were given in a study of a group of 1000 subscribers to a certain magazine: In reference to job, martial status and education, there were 312 professionals, 470 married persons, 525 college graduates, 42 professional college graduates, 147 married college graduates, 86 married professionals, and 25 married professional college graduates. Show that the numbers reported in the study must be incorrect.

27. Suppose 121 nn EEEE is a decreasing sequence of events. Show that

nni i EPEP

lim1 .

28. Let A, B, C be three arbitrary events. Show that

2 CPBPAPCBAP .

29. If 5 married couples are randomly arranged in a row of seats, find the probability that no

couple sits together. 30. To redeem a souvenir from a chain store, one must collect a set of five different types of

coupons, namely A, B, C, D, E Suppose that a customer has collected 12 coupons independently, with each equally likely to be one of these five types. (a) Find the probability that the customer has no coupon of type A. (b) Find the probability that the customer has no coupon of type A and type B. (c) Find the probability that the customer can redeem a souvenir.

31. Twenty people get on an elevator at the ground floor of a building. If each person is equally

likely to choose any of the 10 floors, independently, what is the probability that the elevator stops at all 10 floors?

32. If two fair dice are rolled, what is the condition probability that the first one lands on 6 given

that the sum of the dice is i? Compute for all values of i between 2 and 12. 33. Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given

that the dice land on different numbers? 34. The probabilities that a certain type of cell produces during its lifetime 0,1,2 cells of the first

generation are 1/4 , 5/8, 1/8, and any cell of the first generation produces cells of the second generation under the same conditions. Find the probabilities

(i) that three cells will produce exactly three cells of the first generation? (ii) that two cells will produce at least one cell of the first generation? (iii) that one cell will give rise to at least one cell of the second generation?

35. A simplified model for the movement of the price of a stock supposes that on each day the

stock’s price either moves up 1 unit with probability p or it moves down 1 unit with probability p1 . The changes on different days are assumed to be independent. (a) What is the probability that after 2 days the stock will be at its original price? (b) What is the probability that after 3 days the stock’s price will have increased by 1 unit? (c) Given that after 3 days the stock’s price has increased by 1 unit, what is the probability

that it went up on the first day?

14/15

P. 6

36. An urn contains 4 red and 6 black balls. Players A and B withdraw balls from the urn consecutively until a red ball is selected. The one who drew the red ball is the winner. Find the probability that A will win. (A draws the first ball, then B, and so on. There is no replacement of the drawn balls.)

37. Prove or give counterexamples to the following statements:

(a) If E is independent of F and E is independent of G, then E is independent of GF . (b) If E is independent of F, and E is independent of G, and GF , then E is

independent of GF . (c) If E is independent of F, and F is independent of G, and E is independent of GF , then

G is independent of FE . (d) If E is independent of F, then they are conditionally independent given that G occurs.

38. Urn I contains 3 white and 4 red balls, whereas urn II contains 2 white and 1 red ball. A ball

is randomly chosen from urn I and put into urn II, then a ball is randomly selected from urn II. (a) What is the probability that the ball selected from urn II is white? (b) What is the conditional probability that the transferred ball was white, given that a white

ball is selected from urn II? 39. (a) If 0AP , show that

ABPBABAP || . (b) Describe the inequality in part (a) verbally.

40. (Craps) The dice game “craps” is one of the most popular casino games. In each game we bet on the

pass line by placing our (dollar) chips on it, and the shooter (one of the players with the largest bet) throws two dice. The rules for payoff are as follows: If the sum of numbers on two dice is 7 or 11, the shooter and players who bet with him/her on pass line win; the player loses on the first throw if the shooter gets 2, 3, or 12 points. If the outcome is 4, 5, 6 8, 9 or 10 points on the first throw, the shooter continues to throw the dice repeatedly until he or she produces either a 7 or the first number thrown; in the latter the player wins, in the former the player loses. Examples of throws are as follows

5 8 2 3 4 6 5 (win) 8 4 2 12 10 11 7 (lose)

12 (lose) 7 (win)

(a) Calculate the probability of winning the pass line version of the game of craps with a pair

of fair dice. From this probability can you see why the game of craps is so popular? (b) Calculate the probability that the game will end on the first or second throw.

41. Ninety-seven percent of all babies survive delivery. However, 12 percent of all births involve

Cesarean (C) sections, and when a C-section is performed the baby survives 95 percent of the time. If a randomly chosen pregnant woman does not have a C-section, what is the probability that her baby survives?

14/15

P. 7

42. An auto insurance company classifies its policyholders as good, bad, or average risks: 30% are deemed good risks, 20% are deemed bad risks, and 50% are deemed average risks. Historical data suggest tat 5% of the good risks, 40% of the bad risks, and 10% of the average risks will be involved in an accident in the coming year.

(a) What is the probability that a randomly chosen customer files an accident claim in the

coming year? (b) An accident claim has just been filed with the company. What is the probability that this

customer was classified as a good risk? A bad risk? An average risk? (c) The company would like to have accident claims on at most 10% of its policies.

Consequently, it decides to cancel the policies of some bad risk customers and replace these policies with average risks. Of the company’s customers, 30% will remain classified as good risks. What is the smallest percentage of the company’s customers who must be classified as average risks for the fraction of customers filing accident claims to be at most 10%?

43. A certain college has observed that 20% of its incoming freshmen are unqualified and drop

out within the first 6 months. To better predict a student’s success, the college has decided to administer a test to all freshmen when they first enrol. The college observes over a period of many years that 85% of qualified students had passed the test, and 80% of unqualified students had failed the test. If a freshman passed the test, what is the probability that he/she will be unqualified and drop out within the first 6 months?

44. A broker handles futures contracts on oil, barley, and orange juice. Of the broker’s orders, 60% are for oil futures, 30% are for barley futures, and 10% are for orange juice futures. On a given day, 40% of the orders for oil futures are buys, 55% of the orders for barley futures are sells, and 35% of the orders for orange juice futures are sells. At the end of the day, a clerk notices that one of the sell orders has the name of the commodity future sold omitted. What is the probability that this order was for a future on oil? On barley? On orange juice?

45. Suppose that there was a cancer diagnostic test that was 98 percent accurate both on those that do and those that do not have the disease. If 0.5 percent of the population have cancer, compute the probability that a tested person has cancer, given that his or her test result indicates so.

46. Assume that there are two types of drivers. The less safe drivers, who comprise 20% of the

population, have probability 0.6 each of causing an accident in a year. The rest of the population are safe drivers, who have probability 0.1 each of causing an accident in a year. The insurance premium is $7000 times one’s probability of causing an accident in the coming year. Assume that for each driver, the events of causing an accident in different years are independent. (a) What is the probability that a driver selected at random from the population will cause an

accident in this year? (b) A new subscriber had caused an accident this year. What is the probability that he/she is a

less safe driver? (c) How much should be the premium charged to the subscriber in part (b) for the following

year?

14/15

P. 8

47. A student wishes to pass degree examination in two subjects. He estimates that if he concentrates his revision effort on one of them (Strategy I), he will have probability 0.8 of passing in that subject, but 0.4 of passing in the other. If he revises for both examinations (Strategy II), he has probability 0.6 of passing each. His performances in the two exams will be independent of one another.

If he fails either or both of the examinations, he may resit. If he resits one examination, he will pass with probability 0.9, and if he resits both, he will pass both with probability 0.6. Which of the two revision strategies will give him the best chance of eventually passing in both subjects? If he fails just one of the original examinations, he may take a vacation job, which will reduce his probability of passing the resit. What would the value of this reduced probability have to be to make the probabilities of eventually passing in both subjects under the two strategies equal?

48. Assume that the last juror to be selected for a panel to serve in the trial of a certain accused

person will be 1 of 12 candidates. Of the 12, 5 are young persons and the other 7 are older people, this distribution being fairly representative of the proportions of young and older among jurors in the county generally. A special survey in the county has shown that 25 percent of jurors are sympathetic to a defendant accused of the particular type of offense at issue in the present trial, and it is known that 70 percent of sympathetic jurors are older people. Given that the final juror chosen is an older person, what is the probability that he or she is sympathetic to the accused?

49. One probability class of 30 students contains 16 that are good, 9 that are average, and 5 that

are of poor quality. A second probability class, also of 30 students, contains 2 that are good, 10 that are average, and 18 that are poor. You (the expert) are aware of these numbers, but you have no idea which class is which. If you examine one student selected at random from each class and find that the student from class A is an average student whereas the student from class B is a poor student, what is the probability that class A is the superior class?

50. A small plane has gone down, and the search is organized into three regions. Starting with

the likeliest, they are :

Region Initial Chance Plane is There Chance of Being Overlooked in the Search

Mountains 0.6 0.4 Prairie 0.1 0.2

Sea 0.3 0.8 The last column gives the chance that if the plane is there, it will not be found. For example,

if it went down at sea, there is 80% chance it will have disappeared, or otherwise not be found. Since the pilot is not equipped to long survive a crash in the mountains, it is particularly important to determine the chance that the plane went down in the mountains.

(a) Before any search is started, what is this chance ? (b) The initial search was in the mountains, and the plane was not found. Now what is the

chance the plane is nevertheless in the mountains ? (c) The search was continued over the other two regions, and unfortunately the plane was not

found anywhere. Finally now what is the chance that the plane is in the mountains ? (d) Describing how and why the chances changed from (a) to (b) to (c).

14/15

P. 9

51. (The Monty Hall Problem) Suppose you are on a game show, and you are given the choice of three boxes. In one box is a key to a new BMW while empty in others. You pick a box, say box A. Then the host, Monty Hall, who knows what are inside the boxes, opens another box, say box B, which is empty. He then says to you, “Do you want to abandon your box and pick box C?” Is it a wise decision to trade box A for box C?

52. An economist believes that during periods of high economic growth, the US dollar

appreciates (i.e., becomes more valuable) with probability 0.7; in periods of moderate economic growth, the dollar appreciates with probability 0.4; and during periods of low economic growth, the dollar appreciates with probability 0.2. During any period of time, the probability of high economic growth is 0.3, the probability of moderate economic growth is 0.5, and the probability of low economic growth is 0.2.

(a) Draw a tree diagram to describe all the possible situations. (b) What is the probability that the US dollar is appreciating but the economic grow is not

high. (c) Suppose that the US dollar has been appreciating during the present period. What is the

probability that we are experiencing a period of high economic growth? (d) Suppose that the US dollar has been appreciating during the present period. If the

economic growth status remains unchanged, what is the probability that the US dollar will appreciate in the next period of time? What assumption have to be made in your calculation?

53. Tom and Bob take turns to toss a coin that lands on heads with probability , and the one

who first obtain a head wins the game. Suppose that Tom toss the coin first. Denote Tp as the event that Tom can win the game.

(a) Show that TT pp 21 . (HINT: conditional on the outcomes of first two tosses.) (b) Express the winning probabilities of Tom and Bob in terms of .

54. A and B plays a series of games. Each game is independently won by A with probability 0.6

and by B with probability 0.4. They stop when their total numbers of wins differ by two. Then the player with the greater number of total wins is declared the match winner. Find the probability that A will be the match winner.

55. (Penney’s game) Independent flips of a biased coin that lands on heads with probability 0.7

are made. Each of two players, A and B, had chosen one out of the eight triplet: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} and the player whose triplet occurs first wins. For example, suppose A had chosen HHT and B had chosen THT. Then if the flipped coin shows the sequence HHHT…, A wins; and if the flipped coin shows the sequence TTTHT…, B wins. Since the coin is biased towards heads, the triplet HHH seems to be a good choice. (a) What is the probability that the pattern THH occurs before the pattern HHH? (b) What is the probability that the pattern HTH occurs before the pattern THH? (c) What is the probability that the pattern HHH occurs before the pattern HTH? (d) Comment on the above results.

14/15

P. 10

56. A coin is tossed repeatedly and the outcomes are recorded as a sequence of H’s (heads) and T’s (tails). Define each set of consecutive H’s and each set of consecutive T’s as a run, e.g. the sequence HTTHHTTTTT contains a run of one H, followed by a run of two T’s, then a run of two H’s and finally a run of 4 T’s. Find the probability that a run of r heads is observed before a run of s tails.

57. Let ,...,2,1 nS and suppose that A and B are, independently, equally likely to be any of

the n2 subsets (including the null set and S itself) of S.

(a) Show that n

BAP

4

3.

HINT: Let BN denote the number of elements in B. Use

n

i

iBNPiBNBAPBAP0

|

(b) Show that n

BAP

4

3 .

58. Players are of equal skill, and in a contest the probability is 0.5 that a specified one of the two

contestants will be the victor. A group of n2 players are paired off against each other at random. The 12 n winners are again paired off randomly, and so on, until a single winner remains. Consider two specified contestants, A and B, and define the events iA , ni and E

by iA : A plays in exactly i contests;

E : A and B never play with each other.

(a) Find iAP , ni ,...,2,1 .

(b) Let cn EPP . Show that

1

2

2

1

12

22

12

1

nn

n

nn PP

(c) Using the result in part (b) or otherwise, find EP .

59. (Borel-Cantelli Lemma)

Let ,, 21 EE be a sequence of events and define

1n nj

jEE .

(a) Verbally interpret the meaning of the event E.

(b) Suppose

1jjEP . Show that 0EP . (HINT: continuity property of probability)

(c) Suppose

1jjEP and further that the events ,, 21 EE are independent. Show that

1EP . (HINT: consider cE and you may find the inequality xex 1 useful.)

14/15

P. 11

60. (Inverse Bayes Formula) (a) Let nBBB ,...,, 21 be a partition of the sample space and A be an arbitrary event. Show

that

1

1 |

|

n

k k

k

BAP

ABPAP .

(It is known as the Inverse Bayes Formula, by which the unconditional probability AP

can be evaluated based upon the conditional probabilities ABP k | and kBAP | ,

nk ,...,2,1 .)

(b) The reliability of a particular skin test for tuberculosis (TB) is as follows: if the subject has TB, the test comes back positive 98% of the time. If the subject does not have TB, the test comes back negative 99% of the time. (Another way to say this is that the sensitivity of the test is 0.98, and the specificity of the test is 0.99.)

For a large population, the predictive power of the test is found as follows: out of the persons with positive test results, only 1.9227% was actually found to have TB; out of the persons with negative test results, 99.9996% was actually found to have no TB. (Another way to say this is that the positive predictive probability of the test is 0.019227, and the negative predictive probability of the test is 0.999996.) What is the prevalence of TB, i.e. the proportion of persons who have TB, in this population?

14/15

P. 12

Challenging problems Here are some interesting and challenging problems in probability. Try your best to tackle these problems. There is no absolute right or wrong. You have no need to submit your solution for these problems. However, you are encouraged to think about them and discuss with your classmates. The Secretary Problem (Also known as the Dowry Problem) Suppose an executive needs to hire a new secretary. She finds n possible candidates for the job. Naturally, she wants to hire the best candidate, so she interviews them one at a time. After each interview, she knows only whether that candidate is the best of the ones she has already interviewed. She has no idea how the candidate compares to the remaining candidates. However, she must decide at that time whether or not to hire the candidate. We assume that the candidates come in random order, so that the probability of any sequence of candidates is equally likely. What strategy would you suggest to her, so that she can have the highest chance to hire the best candidate? What is the probability that she can hire the best candidate by using your strategy? Is Probability a Transitive Property? Your good friend Paul stops by one afternoon for a friendly game of cards. Of course, you know Paul well enough to suspect that any time there's a game being played, he'll be ready to make a quick buck at your expense. But this time, he assures you it'll be different, "This game is complete chance, here I'll show you. Got a deck of cards?" You find a set of cards and hand them to him. Paul proceeds to find nine cards: 2, 3, 4, 5, 6, 7, 8, 9, 10. He hands the nine cards to you. "Group them by suit," he says. You do as he instructs- you now have three groups of three cards. "Now flip them face down and shuffle the cards within each pile, but don't mix the piles together." When you've done this, he proposes the game: "Okay, here's how this game is played. You pick a random card from any pile, but don't show me. Then I'll point to one of the other piles, and then you pick a random card from that pile. If your card is higher, I owe you a dollar. If my card is higher, you owe me a dollar. Can't be much fairer than that, can it?" Knowing Paul, it probably could be. But it does sound fair- so what's the catch?

14/15

P. 13

The Prisoners Problem Three prisoners, A, B and C are on death row. The governor decides to pardon one of the three and chooses at random the prisoner to pardon. He informs the warden of his choice but requests that the name be kept secret for a few days. The next day, A tries to get the warden to tell him who had been pardoned. The warden refuses. A then asks which of B or C will be executed. The warden thinks for a while, then tells A that B is to be executed. Warden’s reasoning : Each prisoner has a 1/3 chance of being pardoned. Clearly either B or C must be executed, so I have given A no information about whether A will be pardoned. A’s reasoning : Given that B will be executed, then either A or C will be pardoned. My chance of being pardoned has risen to 1/2. Whose reasoning is correct? Le Truel : A Problem of Competition Once upon a time a king announced that the husband of his lovely daughter would be decided by Le Truel, which was to be a “three-man duel” with pistols to decide which of three competing suitors would win the princess. In a duel, as was the custom in those days, it was winner-take-all; but, for Le Truel the rules had to be amended. If exactly one man survived, he can win the princess. However, if none or more than one survived, the princess would remain forever unwed. Time hasn’t changed everything. As you would expect, the princess was rich and beautiful, so naturally each suitor wished to maximize his own probability of being the sole survivor – if he couldn’t marry the princess, he’d rather be dead. The procedure of Le Truel was conceived by the king’s chancellor. The three competitors would stand at the vertices of an equilateral triangle 25 paces on a side. A total of six bullets would be dispensed, one at a time to the suitors in order. The chancellor established the order alphabetically : Sir Albert, Sir Bertram, Lord Cass. No man would be permitted to fire more than two bullets. If a man no longer lived when his turn came, his bullet would be retained by the referee. Each time a bullet was dispensed, the recipient had complete freedom to choose what he did with it, except that he could not choose to save it for later use. Before the appointed day, both the thoughtless king and chancellor are busy with the arrangement of the truel and hadn’t given a thought to the relative marksmanship of the three suitors. The three young men, on the other hand, could think of little else. All three knew each other’s skills. They knew that the probability of Lord Cass’ killing anyone he aimed at from 25 paces was 1. Cass was a dead shot! Bertram, on the other hand, had a probability of only 3/4. But Albert,, who had spent a good deal of his youth reading rather than improving his marksmanship, had a probability of only 2/3. Furthermore, the time was too short for either Albert or Bertram to improve their chances by practice and Cass didn’t need to. Meanwhile, the excitement mounted. Sir Albert is the poorest marksman, but he gets to go first. What will be the best thing for him to do with his first bullet? What is the probability that the beautiful princess will remain an old maid? Which man has the best chance of winning?

14/15

P. 14

The Lost Boarding Pass One hundred people line up to board an airplane. Each has a boarding pass with assigned seat. However, the first person to board has lost his boarding pass and takes a random seat. After that, each person takes the assigned seat if it is unoccupied, and one of unoccupied seats at random otherwise. What is the probability that the last person to board gets to sit in his assigned seat? The Mystery Thirteenth Floor Mr. Smith works on the 13th floor of a 15 floor building. The only elevator moves continuously through floor 1, 2, …, 15, 14, …, 2, 1, 2, …, except that it stops on a floor on which the button has been pressed. Assume that time spent loading and unloading passengers is very small compared to the travelling time. Mr. Smith complains that at 5pm, when he wants to go home, the elevator almost always goes up when it stops on his floor. What is the explanation? Now assume that the building has n elevators, which move independently. What is the proportion of time the first elevator on Mr. Smith's floor moves up? Guessing the Color : Does a Sample Really Help? John comes to tom, David, Joe with an urn. John says, “Here is an urn containing three balls. The color of each ball can either be black or white. I shall give you one dollar if you can guess correctly the color of the ball you are going to pick from the urn.” Tom says, “It makes no difference to me whether I guess the color, black or white.” David adds, “It will be helpful if we can pick a sample from the urn.” Tom nods his head. Joe is in deep thinking and does not seem completely agree with David. John says without hesitation, “That is all right. You can pick one ball from the urn but you are not allowed to put it back to the urn.” David puts his hand in the urn and randomly picks one ball. The ball is white. He says, “As this ball is white, it is reasonable to believe that the remaining balls are more likely to be white. I guess the color of the next ball drawn is white.” Tom says, “I don’t think so. Now the urn has one white ball less. I guess that the next ball is black.” John turns his head to Joe waiting for his response. Joes says at last, “The color of the ball you pick cannot alter the color of the remaining balls. They are independent. How can we learn anything about the color of the next ball drawn from that sample? Betting on black or white is just the same to me.” Now, which one should we agree? What should be our decision on guessing the color? Does a sample really help?