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    Fifty Challenging Problems in Probability with Solutions
    Fifty Challenging Problems in Probability with Solutions
    Fifty Challenging Problems in Probability with Solutions
    Ebook240 pages2 hoursDover Books on Mathematics

    Fifty Challenging Problems in Probability with Solutions

    By Frederick Mosteller

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    About this ebook

    Can you solve the problem of "The Unfair Subway"?Marvin gets off work at random times between 3 and 5 p.m. His mother lives uptown, his girlfriend downtown. He takes the first subway that comes in either direction and eats dinner with the one he is delivered to. His mother complains that he never comes to see her, but he says she has a 50-50 chance. He has had dinner with her twice in the last 20 working days. Explain.Marvin's adventures in probability are one of the fifty intriguing puzzles that illustrate both elementary ad advanced aspects of probability, each problem designed to challenge the mathematically inclined. From "The Flippant Juror" and "The Prisoner's Dilemma" to "The Cliffhanger" and "The Clumsy Chemist," they provide an ideal supplement for all who enjoy the stimulating fun of mathematics.Professor Frederick Mosteller, who teaches statistics at Harvard University, has chosen the problems for originality, general interest, or because they demonstrate valuable techniques. In addition, the problems are graded as to difficulty and many have considerable stature. Indeed, one has "enlivened the research lives of many excellent mathematicians." Detailed solutions are included. There is every probability you'll need at least a few of them.
    LanguageEnglish
    PublisherDover Publications
    Release dateApr 26, 2012
    ISBN9780486134963
    Fifty Challenging Problems in Probability with Solutions

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    Reviews for Fifty Challenging Problems in Probability with Solutions

    4/5

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    • 3/5

      Dec 27, 2020

      15 years after ending my computer science and mathematical studies, I thought it was time for me to try out this little booklet. I think I made a good start, but soon enough the challenges became too hard for me to solve unguided. :-) Still, it served as a good reminder that statistics are not at all intuitive, which is especially useful to know in this day and age of big data (and COVID-19 related statistics all over the news).

      The solutions were always clear enough however and the collection had no problem whatsoever keeping my interest level high. The reason I'm not 100% sure this publication deserves more stars is that, even though maths never gets old, the problems used to frame the maths do feel quite out of place in the 21st century.
    • 3/5

      May 30, 2015

      Mostly straight-forward questions. Some solutions are merely discussions instead of rigorous proofs. Concluding problems are, let's just say, a bitch.

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    Fifty Challenging Problems in Probability with Solutions - Frederick Mosteller

    Fifty Challenging Problems in Probability

    1. The Sock Drawer

    A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is e9780486134963_i0002.jpg . (a) How small can the number of socks in the drawer be? (b) How small if the number of black socks is even?

    2. Successive Wins

    To encourage Elmer’s promising tennis career, his father offers him a prize if he wins (at least) two tennis sets in a row in a three-set series to be played with his father and the club champion alternately: father-champion-father or champion-father-champion, according to Elmer’s choice. The champion is a better player than Elmer’s father. Which series should Elmer choose?

    3. The Flippant Juror

    e9780486134963_i0003.jpg

    A three-man jury has two members each of whom independently has probability p of making the correct decision and a third member who flips a coin for each decision (majority rules). A one-man jury has probability p of making the correct decision. Which jury has the better probability of making the correct decision?

    4. Trials until First Success

    On the average, how many times must a die be thrown until one gets a 6?

    5. Coin in Square

    In a common carnival game a player tosses a penny from a distance of about 5 feet onto the surface of a table ruled in 1-inch squares. If the penny ( e9780486134963_i0004.jpg inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise he loses his penny. If the penny lands on the table, what is his chance to win?

    6. Chuck-a-Luck

    Chuck-a-Luck is a gambling game often played at carnivals and gambling houses. A player may bet on any one of the numbers 1, 2, 3, 4, 5, 6. Three dice are rolled. If the player’s number appears on one, two, or three of the dice, he receives respectively one, two, or three times his original stake plus his own money back; otherwise he loses his stake. What is the player’s expected loss per unit stake? (Actually the player may distribute stakes on several numbers, but each such stake can be regarded as a separate bet.)

    7. Curing the Compulsive Gambler

    Mr. Brown always bets a dollar on the number 13 at roulette against the advice of Kind Friend. To help cure Mr. Brown of playing roulette, Kind Friend always bets Brown $20 at even money that Brown will be behind at the end of 36 plays. How is the cure working?

    (Most American roulette wheels have 38 equally likely numbers. If the player’s number comes up, he is paid 35 times his stake and gets his original stake back; otherwise he loses his stake.)

    8. Perfect Bridge Hand

    We often read of someone who has been dealt 13 spades at bridge. With a well-shuffled pack of cards, what is the chance that you are dealt a perfect hand (13 of one suit)? (Bridge is played with an ordinary pack of 52 cards, 13 in each of 4 suits, and each of 4 players is dealt 13.)

    9. Craps

    The game of craps, played with two dice, is one of America’s fastest and most popular gambling games. Calculating the odds associated with it is an instructive exercise.

    The rules are these. Only totals for the two dice count. The player throws the dice and wins at once if the total for the first throw is 7 or 11, loses at once if it is 2, 3, or 12. Any other throw is called his point.¹ If the first throw is a point, the player throws the dice repeatedly until he either wins by throwing his point again or loses by throwing 7. What is the player’s chance to win?

    10. An Experiment in Personal Taste for Money

    e9780486134963_i0005.jpg

    (a) An urn contains 10 black balls and 10 white balls, identical except for color. You choose black or white. One ball is drawn at random, and if its color matches your choice, you get $10, otherwise nothing. Write down the maximum amount you are willing to pay to play the game. The game will be played just once.

    (b) A friend of yours has available many black and many white balls, and he puts black and white balls into the urn to suit himself. You choose black or white. A ball is drawn randomly from this urn. Write down the maximum amount you are willing to pay to play this game. The game will be played just once.

    Problems without Structure (11 and 12)

    Olaf Helmer and John Williams of The RAND Corporation have called my attention to a class of problems that they call problems without structure, which nevertheless seem to have probabilistic features, though not in the usual sense.

    11. Silent Cooperation

    Two strangers are separately asked to choose one of the positive whole numbers and advised that if they both choose the same number, they both get a prize. If you were one of these people, what number would you choose?

    12. Quo Vadis?

    Two strangers who have a private recognition signal agree to meet on a certain Thursday at 12 noon in New York City, a town familiar to neither, to discuss an important business deal, but later they discover that they have not chosen a meeting place, and neither can reach the other because both have embarked on trips. If they try nevertheless to meet, where should they go?

    13. The Prisoner’s Dilemma

    Three prisoners, A, B, and C, with apparently equally good records have applied for parole. The parole board has decided to release two of the three, and the prisoners know this but not which two. A warder friend of prisoner A knows who are to be released. Prisoner A realizes that it would be unethical to ask the warder if he, A, is to be released, but thinks of asking for the name of one prisoner other than himself who is to be released. He thinks that before he asks, his chances of release are e9780486134963_i0006.jpg . He thinks that if the warder says "B will be released," his own chances have now gone down to e9780486134963_i0007.jpg , because either A and B or B and C are to be released. And so A decides not to reduce his chances by asking. However, A is mistaken in his calculations. Explain.

    14. Collecting Coupons

    Coupons in cereal boxes are numbered 1 to 5, and a set of one of each is required for a prize. With one coupon per box, how many boxes on the average are required to make a complete set?

    15. The Theater Row

    Eight eligible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a theater. On the average, how many pairs of adjacent seats are ticketed for marriageable couples?

    16. Will Second-Best Be Runner-Up?

    A tennis tournament has 8 players. The number a player draws from a hat decides his first-round rung in the tournament ladder. See diagram.

    e9780486134963_i0008.jpg

    Tennis tournament ladder of 8.

    Suppose that the best player always defeats the next best and that the latter always defeats all the rest. The loser of the finals gets the runner-up cup. What is the chance that the second-best player wins the runner-up cup?

    e9780486134963_i0009.jpg

    17. Twin Knights

    (a) Suppose King Arthur holds a jousting tournament where the jousts are in pairs as in a tennis tournament. See Problem 16 for tournament ladder. The 8 knights in the tournament are evenly matched, and they include the twin knights Balin and Balan.² What is the chance that the twins meet in a match during the tournament?

    (b) Replace 8 by 2n in the above problem. Now what is the chance that they meet?

    18. An Even Split at Coin Tossing

    When 100 coins are tossed, what is the probability that exactly 50 are heads?

    19. Isaac Newton Helps Samuel Pepys

    Pepys wrote Newton to ask which of three events is more likely: that a person get (a) at least 1 six when 6 dice are rolled, (b) at least 2 sixes when 12 dice are rolled, or (c) at least 3 sixes when 18 dice are rolled. What is the answer?

    20. The Three-Cornered Duel

    A, B, and C are to fight a three-cornered pistol duel. All know that A’s chance of hitting his target is 0.3, C’s is 0.5, and B never misses. They are to fire at their choice of target in succession in the order A, B, C, cyclically (but a hit man loses further turns and is no longer shot at)

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