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Challenging Problems
Simplifying 50 Challenging Problems
| Question | Answer |
|---|---|
| THE SOCK DRAWER: The probability that both are red is 0.5. (a) number of socks in the drawer be (b) black socks is even | (a) 4 (b) 21 |
| SUCCESSIVE WINS: To encourage Elmer's promising tennis career. If he wins (at least) two tennis sets in a row in a three-set series. | Champion-Father-Champion or CFC |
| THE FLIPPANT JUROR: Majority rules. Better probability of making the correct decision. | One man jury has the same probability of making the correct decision with the three-man jury. |
| TRIALS UNTIL FIRST SUCCESS: Average. A die be thrown until one gets a 6. | 6 times |
| COIN IN SQUARE: Tosses a penny from a distance of about 5 feet onto the surface of a table ruled in I-inch squares. | 9/256 or less than 1/28 |
| CHUCK-A-LUCK: His original stake plus his own money back; otherwise he loses his stake. What is the player's expected loss per unit stake? | 0.079 or almost 8% |
| CURING THE COMPULSIVE GAMBLER: Number 13 at roulette against the advice of Kind Friend. How is the cure working? | +2.79 dollars per 36 trials |
| PERFECT BRIDGE HAND: Dealt 13 spades at bridge. Chance that you are dealt a perfect hand. | 4 x (13!39!/52!) or 4 x (n!(52-n)!/52!) Where in this case, n=13 |
| CRAPS: The game of craps. What is the player’s chance to win? | 0.49293 |
| AN EXPERIMENT IN PERSONAL TASTE FOR MONEY: An urn contains 10 black balls and 10 white balls. Write down the maximum amount you are willing to pay to play the game. | Subjective question. No one can say what amount is appropriate for you to pay for either game. Expected value is half of the bet. Thus, betting $10 would give us $5 as expected value because of 50-50 chance of winning (random probability). |
| SILENT COOPERATION: Two strangers are separately asked to choose one positive whole numbers. | 1 (natural choice) 3 (popular choice) 7 (popular choice) |
| QUO VADIS?: Two strangers who have a private recognition signal. If they try nevertheless to meet, where should they go? | "Empire State Building" or for easier choice "San Francisco or Paris" |
| THE PRISONER'S DILEMMA: Three prisoners, A, B, and C, with apparently equally good records have applied for parole. One prisoner other than himself. | 0.67 or 2/3 |
| COLLECTING COUPONS: How many boxes on the average are required to make a one complete set? | 11.43 Boxes |
| THE THEATER ROW: Eight eligible bachelors and seven beautiful models. | 7.47 or 3.27 |
| WILL SECOND-BEST BE RUNNER-UP?: A tennis tournament has 8 players. Chance that the second best player wins the runner-up cup? | P = 0.501 |
| TWIN KNIGHTS: Jousting tournament. (a) What is the chance that the twins meet in a match during the tournament? (b) Replace 8 by 2n | (a) 1/4 (b) 1/(2^(n-1)) |
| AN EVEN SPLIT AT COIN TOSSING: Exactly 50 are heads. | 0.08 |
| ISAAC NEWTON HELPS SAMUEL PEPYS: Pepys wrote Newton to ask which of three events is more likely. | At least 1 six when At least 1 six when 6 dices n 6 dices are rolled. |
| THE THREE-CORNERED DUEL: B never misses. | Allow A to undeniably miss. |
| SHOULD YOU SAMPLE WITH OR WITHOUT REPLACEMENT?: Two urns contain red and black balls, all alike except for color. How do you order the second drawing? | 5/8, Without Replacement. |
| THE BALLOT BOX: Atleast once after the first tally the candidates have the same number of tallies? | 8/10 |
| TIES IN MATCHING PENNIES: Chance that at no time during the game will they be even? | (N/n)/2^N |
| THE UNFAIR SUBWAY: Had dinner with her twice in the last 20 working days. | To go uptown, Marvin must arrive in the 1-minute interval between a downtown and an uptown train. |
| LENGTHS OF RANDOM CHORDS: Lengths exceeds the radius of the circle? | 2/3 |
| THE HURRIED DUELERS: Fraction of duels lead to violence? | 1/6 |
| CATCHING THE CAUTIOUS COUNTERFEITER: (a) Minter’s peculations go undetected? (b) both IOO's are replaced by n? | (a) 0.366 (b) P = (1 - 1/n)^n |
| CATCHING THE GREEDY COUNTERFEITER: Chance that the sample of n coins contains exactly r false ones? | e^m |
| MOLDY GELATIN: Fraction of plates has exactly 3 colonies? | 2^n=(0.4/(√m)) |
| EVERYTHING THE SALES: Chance that he sells an even number of cakes? | 0.568 |
| BIRTHDAY PAIRINGS: Exceeds ½ that two or more of them have the same birthday? | 23 |
| FINDING YOUR BIRTHMATE: Birthdays you need to ask about to have a 50-50 chance? | 253 |
| RELATING THE BIRTHDAY PAIRINGS AND BIRTHMATE PROBLEMS: What should n be in the personal birthmate problem to make your probability of success approximately PR. | (r(r-1))/2 |
| BIRTHDAY HOLIDAYS: Expected total number of man-days worked per year in a factory. | 364 |
| THE CLIFF-HANGER: Drunken man over the edge. | Probability of Disaster = 107/243 Probability of Escaping(Infinite Steps) = 1/2 |
| GAMBLER'S RUIN: They play until one is bankrupt. | 1/3 |
| BOLD PLAY VS CAUTIOUS PLAY: Compare the merits of the strategies. | 0.474 > 0.11, bet all at 0.474 > 0.11, bet all at once. |
| THETHICK COIN: A one-third chance of landing on edge? | 0.354 of the Diameter. |
| THE CLUMSY CHEMIST: Average length of the fragment with the blue dot? | 3 inches |
| THE FIRST ACE: How many cards are required to produce FIRST ACE. | 10.6 cards on the average. |
| THE LOCOMOTIVE PROBLEM: (a) One day you see a locomotive and its number is 60 (b) 5 locomotives and largest number is observed is 60 | (a) 119 (b) 71 (Estimation only since there is no right answer for the question) |
| THE LITTLE END OF THE STICK: (a) Average length of the smaller piece? (b) Average ratio of smaller length to larger length? | (a) 1/4 L (b) 0.386 |
| THE BROCKEN BAR: Average size of the smallest, of the middle-sized, and the largest pieces. | • 2 • 5 • 11 |
| WINNING AN UNFAIR GAME: Choose in advance the number of plays. | 10 |
| AVERAGE NUMBER OF MATCHES: (a) Card above and the card below in repetitions of this experiment? (b) Letters are put into their own envelopes? | (a) 1 (b) 1 |
| PROBABILITIES OF MATCHES: Probability of exactly r matches. | 0.368 |
| CHOOSING THE LARGEST DOWRY: How should the wise man make his decision? | Pass 37 and take the first candidate thereafter. |
| CHOOSING THE LARGEST RANDOM NUMBER: The king wants the wise man to choose the largest number from among 100. Slips are randomly drawn from the numbers from 0 to 1. | 3/4 or 0.684 0.580 for very large value of n |
| DOUBLING YOUR ACCURACY: Estimate the lengths of two cylindrical rods, one clearly longer than the other. | σ^2/(4/3) |
| RANDOM QUADRATIC EQUATIONS: Probability that the quadratic equation has real roots? | 1! |
| TWO-DIMENTIONAL RANDOM WALK: Chance that the particle returns to the origin? | 1 |
| THREE-DIMENTIONAL RANDOM WALK: Every move the particle has a 50-50 chance of moving one unit up or down. | 0.239 |
| BUFFON'S NEEDLE: When it comes to rest it crosses the line? | 2L/πa |
| BUFFON'S NEEDLE WITH HORIZONTAL AND VERTICAL RULINGS: Mean number of lines the number crosses. | 4/π |
| LONG NEEDLES: Needle be of arbitrary length. | 4L/π |
| MOLINA'S URNS: Drawing from the second is either all whites or all blacks. | 3 ≥ n ≥ 2000 |
Created by:
johneilfeca
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