PROBABILITY SUMMARY

Question 1

Problem:
A drawer contains red socks and black socks. When two socks are drawn at random, the probability that both are red is 1/2.

Questions:
(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

4
21 (Black)

Question 2

Problem:
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.

Question: Which series should Elmer choose?

Answer 2

Champion-father-Champion

Question 3

Problem:
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.

Question: Which jury has the better probability of making the correct decision?

Answer 3

Same Juries (One man and three man jury)

Question 4

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

Answer 4

6

Question 5

Problem:
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.

Question: If the penny lands on the table, what is his chance to win?

Answer 5

9/256
or less than 1/28

Question 6

Problem:
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, 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.

Question: What is the player’s expected loss per unit stake?

Answer 6

8%

Question 7

Problem:
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.

Question: How is the cure working?

Answer 7

Gain of 2.79 Dollars per 36 trials

Question 8

We often read of someone who has been dealt 13 spades at bridge.

Question: With a well-shuffled pack of cards, what is the chance that you are dealt a perfect hand (13 of one suit)?

Answer 8

6.2999 x 10 ^12

Question 9

Problem:
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.
Question: What is the player’s chance to win?

Answer 9

0.49293

Question 10

Problem:
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.

Question: If you were one of these people, what number would you choose?

Answer 10

1,3,7

Question 11

Problem:
Coupons in cereal boxes are numbered 1 to 5, and a set of one of each is required for a prize.

Question: With one coupon per box, how many boxes on the average are required to make a complete set?

Answer 11

11.43 box

Question 12

Problem:
Eight eligible bachelors and seven beautiful models happen randomly to have purchased single seats in the same 15-seat row of a theater.

Question: On the average, how many pairs of adjacent seats are ticketed for marriageable couples?

Answer 12

7 7/17 or 4.47

Question 13

Problem:
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 below).
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.

Question: What is the chance that the second-best player wins the runner-up cup?

Answer 13

4/7

Question 14

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

Answer 14

0.07959

Question 15

Problem:
Suppose King Arthur holds a jousting tournament where the jousts are in pairs as in a tennis tournament (see diagram below for the tournament ladder).
The 8 knights in the tournament are evenly matched, and they include the twin knights Balin and Balan.

Questions:

(a) 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 15

1/4

Question 15

Problem:
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.

Question: What is the answer?

Answer 15

0.664
0.619
0.597

Question 16

Problem:
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.

Question: What should A’s strategy be?

Answer 16

Miss

Question 17

Problem:
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.

Question: How do you order the second drawing, and how do you decide on the urn?

Answer 17

5/8 (w/o replacement)
21.5/36 (w/ replacement)

Question 18

Problem:
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.

Question: 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 18

8/10

Question 18

Problem:
Players A and B match N times. They keep a tally of their gains and losses.

Question: 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

Problem:
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

2/3 (two points chosen at circumference)
0.866 (distance of the chord from the center)
0.75 (midpoint of the chord is evenly distributed)

Question 20

Problem:
Duels in the town of Discretion are rarely fatal. There, each contestant comes at a random moment between 5 AM and 6 AM on the appointed day and leaves exactly 5 minutes later, honor served, unless his opponent arrives within the time interval and then they fight.

Question: What fraction of duels lead to violence?

Answer 20

23/144 (best answer)
1/6 (approximated)

Question 21

Problem:
(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?

Answer 21

0.366 (100 to a box)

(1- 1/100)^100

Question 22

Problem:
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.

Question: What is the chance that the sample of n coins contains exactly r false ones?

Answer 22

1/√2πm
1/0.4√m

Question 23

Problem: Airborne spores produce tiny mold colonies on gelatin plates in a laboratory. The many plates average 3 colonies per plate. Questions: (1) What fraction of plates has exactly 3 colonies? (2) If the average is a large integer m, what fraction of plates has exactly m colonies?

Answer 23

0.568

Question 24

Problem: A bread salesman sells on the average 20 cakes on a round of his route. Question: What is the chance that he sells an even number of cakes?

Answer 24

23

Question 25

Problem: What is the least number of persons required if the probability exceeds 1/2 that two or more of them have the same birthday?

Answer 25

253

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 26

The Clift-Hanger 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 3, of a step toward the cliff $. What is his chance of escaping the cliff?

Answer 26

107/243 (probability of disaster) 1/2 (probability of escaping)

Question 27

Gambler's Ruin 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 & of the plays. They play until one is bankrupt. What is the chance that Player M wins?

Answer 27

1/3

Question 28

Bold Play ve. Cautious Play 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 28

18/38 > 7.23/57.7 bet all at

Question 29

The Thick Coin How thick should a coin be to have a f chance of landing on edge?

Answer 29

0.354r

Question 30

The Clumsy Chemist 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 30

3 inches

Question 31

The First Ace 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 31

10.6^1.11 card

Question 32

The Locomotive Problem (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.

Answer 32

110 71

Question 33

The Little End of the Stick (a) If a stick is broken in two at random, what is the average length of the smaller piece? (6) (For calculus students.) What is the average ratio of the smaller length to the larger?

Answer 33

1/4 L 0.386

Question 34

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

Answer 34

2,5,11

Question 35

Winning an Unfair Game A game consists of a sequence of plays; on each play either you or your 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 35

10 plays

Question 36

Average Number of Matches The following are two versions of the 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 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 experi-ment? (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?

Answer 36

1 0.368

Question 36

What is the probability that the quadratic equation x^2+2bx+c=0 has real roots?

Answer 36

1

Question 37

Random Walk 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 move is repeated from the new position and so on indefinitely. What is the chance that the particle returns to the origin?

Answer 37

1

Question 38

As in the two-dimensional walk, a particle starts at an origin O 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 north of south

Answer 38

0.239

Question 39

Buffon's Needle A table of infinite expanse has inscribed on it a set of parallel lines spaced 2a units apart. what is the probability that when it comes to rest it crosses a line?

Answer 39

between 0 and π/2

Question 40

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? what is the mean number of crosses?

Answer 40

4/π 4L/π

Question 41

Two urns contain the same total numbers of balls, some blacks and some whites 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 whites or all black

Answer 41

n<2000