MOSTELLER'S FIFTY CHALLENGING PROBLEMS IN PROBABILITY (finalize)

Question 1

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 socks is even?

Answer 1

a. 4
b. 21

Question 2

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?

Answer 2

choose champion-father-champion with 0.512 rather than the father-champion-father with 0.384

Question 3

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 a probability p of making the correct decision. Which jury has the better probability of making the correct decision?

Answer 3

both the jury have the same probability

Question 4

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

Answer 4

6

Question 5

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 loses his penny.

a. If the penny lands on the table, what is his chance to win?

b. If the 1-inch square is made smaller by thickening the lines to 1/16 in wide, what is his chance to win?

Answer 5

a. 1/16
b. 1/28

Question 6

Chuck-a-Luck is a gambling game often played at carnivals and gabling 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.)

Answer 6

17/216 or 0.079

Question 7

Mr. Brown always bets a dollar on the number 13 at a 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.)

Answer 7

Not working, Mr. Brown gains +4.68 - 1.89 = +2.79 dollars per 36 trials

Question 8

We often read of someone wo 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.)

Answer 8

4 × 13!39!/52!

Question 9

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.” 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?

Answer 9

0.27071

Question 10

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 2/3. He thinks that if the warder says, “B will be released,” his own chances 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 chances by asking. However, A is mistaken in his calculations Explain.

Answer 10

2/3

Question 11

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 average are required to make a complete set?

Answer 11

137/12 or 11.42

Question 12

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?

Answer 12

8/15

Question 13

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

Answer 13

4/7

Question 14

(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. 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 what is the chance that they meet?

Answer 14

1/4,1/2^(n-1)

Question 15

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?

Answer 15

2b/(a+b)=4/5

Question 16

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?

Answer 16

At least 1 six when 6 dice are rolled

Question 17

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 of 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?

Answer 17

A does not shoot

Question 18

Players A and B match pennies N times. They keep a tally of their gains and losses. After the first toss, what is the chance that at no time during the game will they be even.

Answer 18

(N n)/2^N

Question 19

If a chord is selected at random on a fixed circle what is the probability that its length exceeds the radius of the circle?

Answer 19

No definite answer: 0.866, 0.75, 0.667

Question 20

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 lead to violence?

Answer 20

1/6

Question 21

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?

Answer 21

0.366

Question 22

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?

Answer 22

1/2

Question 23

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.)

Answer 23

23

Question 24

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?

Answer 24

253

Question 25

If r persons compare birthdays in the pairings 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 success approximately PR?

Answer 25

n=(r(r-1))/2

Question 26

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?

Answer 26

364

Question 27

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, if a step toward the cliff 1/3, What is his chance of escaping the cliff?

Answer 27

4/7

Question 28

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?

Answer 28

4/7

Question 29

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.

Answer 29

$20 at once: 0.474
$1 at a time: 0.110

Question 30

How thick should a coin be to have a 1/3 chance of landing on edge?

Answer 30

0.354r

Question 31

In a laboratory, each of a handful of thin 9-inch glass rods 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 the blue dot?

Answer 31

3 in

Question 32

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?

Answer 32

10.6th card

Question 33

(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 locomotive and the largest number observed is 60. Again, guess how many locomotives the company has.

Answer 33

119, 71

Question 34

(a) If a stick is broken in two at random, what is the average length of the smaller piece?
(b) What is the average ratio of the smaller length to the larger?

Answer 34

1/4, 0.386

Question 35

A bar is broken at random in two places. Find the average size of the smallest, of the middle-sized, and the largest pieces.

Answer 35

1/9, 5/18, 11/18

Question 36

A game consists of a sequence of plays; on each play either you or your opponent scores a point, you with p (less than 1/2), he with probability of 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?

Answer 36

10

Question 37

(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 cards 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 of letters and envelopes to n different persons. The letters are randomly put into envelopes. On the average, how many letters are put into their own envelopes?

Answer 37

1, 1

Question 38

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?

Answer 38

Pass 37 and take the first candidate thereafter

Question 39

What is the probability that the quadratic equation x² + 2bc + c = 0 has real roots?

Answer 39

1

Question 40

Starting from an origin 0, 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 move is repeated from the new position and so on indefinitely. What is the chance that the particle returns to the origin?

Answer 40

1

Question 41

A particle starts at an origin 0 in 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.

Answer 41

0.239

Question 42

A table of infinite expanse has inscribed on it a set of parallel lines spaced 2a units apart. A needle of length 2l (smaller than 2a) is twirled and tossed on the table. What is the probability that when it comes to rest it crosses a line?

Answer 42

2l/πa

Question 43

Suppose we toss a needle of length 21 (less than I) on a grid with both horizontal and vertical rulings spaced one unit apart. What is the mean number of lines the needle crosses?

Answer 43

4l/π