50 Challenging Problems in Probability by Frederick Mosteller
Thickness = 35% Diameter
How thick should a coin be to have a 1/3 chance of landing on edge?
(a) 1/4 (b) 1/((2^n)-1)
(a) Suppose King Arthur holds a jousting tournament where the jousts are in pairs as in a tennis tournament. The 8 Knights in the tournament are evenly matched, and they jnclude the twin knights Balin and Balan. What is the chance that the twins meet in a match during the tournament? (b) Replace 8 by 2^n in the above problem. Now is the chance that they meet?
P=1/6
(a) The king's minter boxes his coins 100 to a box. In each box he puts 1 false coin. The king suspects the minter and from each of 100 boxes draws a random coin and has it tested. What is the chance the minter's peculations go undetected? (b) What if both 100's are replaced by n?
0.4/m
Airborne spores produce tiny mold colonies on gelatin plates in a laboratory. The many plates average 3 colonies per plate. What fraction of plates has exactly 3 colonies? If the average is a large integer m, what fraction of plates has exactly m colonies?
Gain of $2.79 per 36 trials
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 get his original stake back; otherwise he loses his stake.)
10.6th card
Shuffle an ordinary deck of 52 playing cards containing four aces. Then turn up cards from the top until the first ace appears. On the average, how many cards are required to produce the first ace?
3 and 7
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?
mean number of trials = 6 times
on the average, how many times must a die be thrown until one gets a 6?
a. 119 locomotives b. 71 locomotives
(a) A railroad numbers its locomotives in order, 1,2,....,N. One day you see a locomotive and its number is 60. Guess how many locomotives the company has. (b) You have looked at 5 locomotives and the largest number observed is 60. Again guess how many locomotives the company has.
Atleast as much to play the second game as the first
(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 the 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 e played just once.
a. 1/4 of Length b. L=0.386
(a) If a stick is broken in two random, what is the average length of the smaller piece? (b) (For calculus students.) What is the average ratio of the smaller length to the larger?
Smallest = 1/9 Middle sized = 5/18 Large = 11/16
A bar is broken at a random two places. Find the average size of the smallest, of the middle-sized, and of the largest pieces.
P=0.568
A bread salesman sells on the average 20 cakes on a round of his route. What is the chance that he sells an even number of cakes? ( We assume the sales follow the Poisson distribution.)
A. n=4 B. n=21
A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2. (a) How small can the number of socks in the drawer be? (b) How small if the number of black sock is even?
n=10 plays
A game consists of a sequence of plays; on each play either you or your opponent scores a point, you with probability p (less than 1/2), he with probability 1-p. The number of plays is to be even-2 or 4 or 6 and so on. To win the game you must get more than half the points. You know p, say 0.45, and you get a prize if you win. You get to choose in advance the number of plays. How many do you choose?
P=2 (length of needle) / circumference of circle of radius a
A table of infinite expanse has inscribed on it a set of parallel lines spaced 2a units apart. A needle of length 2/ (smaller than 2a) is twirled and tossed on the table. What is the probability that when it comes to rest it crosses a line?
P=4/7
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) 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?
Two-man jury have the same chance of correct decision
A three-man jury has two members each of whom independently has a 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?
Allow A to miss his first shot
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) until only one man is left unhit. What should A's strategy be?
Yes. :)
An unbiased instrument for measuring distances makes random errors whose distribution has standard deviation. You are allowed two measurements all told to estimate the lengths of two cylindrical rods, one clearly longer than the other. Can you do better than to take one measurement on each rod? (An unbiased instrument is one that on the average gives the true measure.)
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As a second task, the king wants the wise man to choose the largest number from among 100, by the same rules, but this time the numbers on the slips are randomly drawn from the numbers from 0 to 1 (random numbers, uniformly distributed.) Now what should the wise man's strategy be?
P=0.239
As in the two-dimensional walk, a particle starts at an origin O in the three-space. Think of the origin as centered in a cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight corners of the cube. Thus, at every move the particle has a 50-50 chance of moving one unit up or down, one unit east or west, and one unit north or south. If the walk continues forever, find the fraction of particles that return to the origin.
Bold Play=0.474 Cautious Play=0.11
At Las Vegas, a man with $20 needs $40, but he is too embarrassed to wire his wife for more money. He decides to invest in roulette (which he doesn't enjoy playing) and is considering two strategies: bet the $20 on "evens" all at once and quit if he wins or loses, or bet on "evens" one dollar at a time until he has won or lost $20. Compare the merits of the strategies.
Expected loss= 8% per play (approximately)
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 stake can be regarded as a separate bet.)
11.42 boxes
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?
P=1/6
Duels in the town of Discretion are rarely fatal. There, each contestant comes at a random moment between 5 A.M. and 6 A.M. on the appointed day and leaves exactly 5 minutes later, honor served, unless his opponent arrives within the time interval and then they fight. What fraction of duels load to violence?
7 7/15 or 7.467
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 man pairs of adjacent seats are ticketed for marriageable couples?
P=1/2
From where he stands, one step toward the cliff would send the drunken man over the edge. He takes random steps, either toward or away from the cliff. At any step his probability of taking a step away is 2/3, of a step toward the cliff is 1/3. What is his chance of escaping the cliff?
P=2/3
If a chord is selected at a random on a fixed circle, what is the probability that its length exceeds the radius of the circle?
(r(r-1))/2
If r persons compare birthdays in the pairing problem, the probability is Pr that at least 2 have the same birthday. What should n be in the personal birthmate problem to make your probability of success approximately Pr?
Probability of winning = 1/16
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 (3/4 inch in diameter) falls entirely inside a square, the player receives 5 cents but does not get his penny back; otherwise he losses his penny. If the penny land on the table, what is his chance to win?
L=3 inches
In a laboratory, each of a handful of thin 9 inch glass rod had one tip marked with a blue dot and the other with a red. When the laboratory assistant tripped and dropped them onto the concrete floor, many broke into three pieces. For these, what was the average length of the fragment with blue dot?
P=8/10
In an election, two candidates, Albert and Benjamin, have in a ballot box a and b votes respectively, a>b, for example, 3 and 2. If ballots are randomly drawn and tallied, what is the chance that at least once after the first tally the candidates have the same number of tallies?
(4*Length)/(no. of crosses or no. of trials)
In the previous problem let the needle be of arbitrary length, then what is the mean number of crosses?
(r(r-1))/2
Labor laws in Erewhon require factory owners to give every worker a holiday whenever one of them has a birthday and to hire without discrimination on grounds of birthdays. Except for these holidays they work a 365-day year. The owners want to maximize the expected total number of man-days worked per year in a factory. How many workers do factories have in Erewhon?
Marvin must arrive in the 1 min interval between a downtown and an uptown train.
Marvin gets off work at a random times between 3 and 5 P.M. His mother lives uptown, his girl friend downtown. He takes the first subway that comes in either direction and eats dinner with the one he is first 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.
At least 1 six when 6 dice are rolled
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?
(N C n)/ 2^n
Player A and B match pennies N times. They keep a tally of their gains and losses. After the first loss, what is the chance that at no time during the game will they be even?
P=1/3
Player M has $1, and Player N has $2. Each play gives one of the players $1 from the other. Player M is enough better than Player N that he wins 2/3 of the plays. They play until one is bankrupt. What is the chance that Player M wins?
P=1
Starting from an origin O, a particle has a 50-50 chance of moving 1 step north or 1 step south, and also a 50-50 chance of moving 1 step east or 1 step west. After the step is taken, the movie is repeated from the new position and so on indefinitely. What is the chance that the particle returns to that origin? Part of lattice of points traveled by the particles in the two-dimensional random walk problem. At each move the particle goes one step north east, northwest, southeast, or southwest from its current position, the directions being equally likely.
Mean = 1.27
Suppose we toss a needle of length 2l (less than 1) on a grid with both horizontal and vertical rulings spaced one unit apart. What is the mean number of lines the needle crosses? (I have dropped the 2a for the spacing because we might as well think of the length of the needle as measured in units of spacing.)
a. 1 match b. 1 letter
The following are two versions of matching problem: (a) From a shuffled deck, cards are laid out on a table one at a time, face up from left to right, and then another deck is laid out so that each of its card is beneath a card of the first deck. What is the average number of matches of the card above and the card below in repetitions of this experiment? (b) A typist types letters and envelopes to n different persons. The letters are randomly put into the envelopes. On the average, how many letters are put into their own envelopes?
P=3/9
The game of craps, played with two dice, is on of America's fastest and most popular gambling games. Calculating the odds associated with it is an inclusive exercise. The rules are these. Only totals for the two dice count. The player throws that dice and winds at once if the total for the first throw is 7 or 11, loses at once if 2,3, or 12. Any other throw is called his "point". 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?
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The king's minter boxes his coins n to a box. Each box contains m false coins. The king suspects the minter and randomly draws 1 coin from each of n boxes and has these tested. What is the chance that the sample of n coins contains exactly r false ones?
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The king, to test a candidate for the position of wise man, offers him a chance to marry the young lady in the court with the largest dowry. The amounts of the dowries are written on slips of paper and mixed. A slip is drawn at random and the wise man must decide whether that is the largest dowry or not. If he decides it is, he gets the lady and her dowry if he is correct; otherwise he gets nothing. If he decides against the amount written on the first slip, he must choose or refuse the next slip, and so on until he chooses one or else the slips are exhausted. In all, 100 attractive young ladies participate, each with a different dowry. How should the wise man make his decision?
P=2/3
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 about 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 2/3. He thinks that if the warder says "B will be released," his own chance have now gone down to 1/2, because either A and B or B and C are to be released. And so A decides not to reduce his chance by asking. However, A is mistaken in his calculation.
Champion - Father - Champion
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?
Empire state building
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?
Draw without replacement with a P=5/8
Two urns contain red and black balls, all alike except for color. Urn A has 2 reds and 1 black, and Urn B has 101 reds and 100 blacks. An urn is chosen at random, and you win a prize if you correctly name the urn on the basis of the evidence of two balls drawn from it. After the first ball is drawn, and its color reported, you can decide whether or not the ball shall be replaced before the second drawing. How do you order the second drawing, and how do you decide on the urn?
z^n=x^n+y^n, z=1st white, y=2nd white, z=3rd white
Two urns contain the same total number of balls, some blacks and some white in each. From each urn are drawn n (>3) balls with replacement. Find the number of drawings and the composition of the two urns so that the probability that all white balls are drawn from the first urn is equal to the probability that the drawing from the second is either all white or all blacks.
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Under the conditions of the previous matching problem, what is the probability of exactly r matches?
P = 4*((13!*39!)/52!) P = 6.299x10^-12
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 card, 13 in each of 4 suits, and each of 4 players is dealt 13.)
n=23 persons
What is the least number of persons required if the probability exceeds 1/2 that two or more of them have the same birthday? (Year of birth need not match.)
P=1/6
What is the probability that the quadratic equation x^2 +2bx+c=0 has real roots?
P=0.08
When 100 coins are tossed, what is the probability that exactly 50 are heads?
n=253 strangers
You want to find someone whose birthday matches yours. What is the least number of strangers whose birthdays you need to ask about to have a 50-50 chance?
