At a Shanghai dart tournament, Henry asked a referee about the rounds played by Jack and Jill.
“I don’t know exactly how many rounds they played before they left, but they flipped a coin each time to see who would go first, and I know that Jack won the toss four times,” said the referee.
“Thanks,” said Henry and pulled out a small notebook and a pen from his shoulder bag. “I’ll work it out.”
How many rounds did Jack and Jill play?
“Master, what are relative primes,” asked Lanoo.
“Lanoo, they are two integers that cannot be divided by any one prime number. The only common divisor of the two numbers is 1,” replied the Master.
“Does that mean that if I have two different integers, no prime number will divide both of them,” repeated Lanoo.
“Yes, indeed,” smiled the Master. “You might try working out the probability that any two random integers less than 100 will be relatively prime,” challenged the Master.
“I will certainly give it a try,” said Lanoo eagerly, pulling out a notebook and a pencil.
Can you help Lanoo work this out? Could the probability you obtain also apply in some way to any two integers greater than 100 picked at random?
King Midas suspected that his minter of gold coins, Horatio, was stealing from the mint by inserting false coins.
As he had 1000 boxes, each containing 100 coins, ready for shipment to King Neptune to pay for construction of some luxury ships, he wondered how he best could check on his suspicions.
So he called on the court mathematician, Leonard, for advice.
“I want to make 99% sure that there is a maximum of 1% of counterfeit coins in the shipment. How can this be done,” he demanded of Leonard.
“Sire, I can have a number of the boxes checked, selecting a sample of one coin at random until we find a false coin,” said Leonard.
“How many boxes will have to be checked,” asked King Midas.
“It depends on the percentage of false coins in them,” replied Leonard. “I will give you a number of boxes to be checked for the case of 1% false coins and a number for 10% false coins.”
After some thought, Leonard gave King Midas the respective number of boxes.
“Very well, proceed. If Horatio has been greedy, he will very quickly lose his head,” thundered King Midas.
How many boxes would have to be checked to ensure that 99% of the shipment had less than one false coin per hundred coins?
If there actually were 10% of false coins, after how many random boxes checked on the average would this be discovered?
Jack and Jill were sitting in the shade of a large tree, relaxing after attending some university lectures.
“What about flipping some coins, Jack,” suggested Jill.
“Good idea, I have a silver dollar,” offered Jack.
“Professor Wiseman suggested a coin flipping game,” said Jill. “We alternately flip a coin and the one who gets the same face as the last face to appear wins. Sound good to you?” asked Jill.
“Sure,” said Jack,“so long as I go first.”
“No problem, Jack, as always, you go first, even after I win,” smiled Jill cheerfully.
They proceeded to play quite a number of games, in which to Jack’s dismay Jill always remained on top.
What is the probability of a game finishing before the coin is flipped six times?
Why did Jill win twice as many times as Jack?
Twelve captured soldiers were presented to Tamerlane, the Turco-Mongol conqueror, in his palace.
“I feel generous today as it is the birthday of my favorite concubine, Lalila, and we have killed enough people this week and built sufficient skull pyramids. You all fought bravely against your capture.”
“Each of you will be blindfolded and ordered to throw your round shield onto the extensive checkered floor. If your shield does not fall completely into a square you will be offered a position in my army, otherwise you will be thrown to the lions,” Tamerlane said with a generous smile.
“By the way,” Tamerlane grinned, “to facilitate your considerations, my court mathematician has measured that precisely four of your round shields can be placed side by side to make up the length of three squares of the checker pattern on the floor. I will give you some moments to think about this, brave soldiers,” he added cheerfully.
The soldiers quickly got busy discussing their fate, one talking about mind over matter.
About how many soldiers would you bet even money on not winding up in the lion pits?
On a winter’s evening, two couples, Jack and Jill, Willy and Lola, went to the famous French restaurant Chez Clouseau in New York to delight their taste buds with a three star Michelin cuisine.
After entering the plush lobby, they had to check their coats in the wardrobe, which was attended by Molly, a distrait blonde.
“Welcome to Chez Clouseau, y’all. Please check in your coats, y’all” said Molly, gracefully extending her arm, fingers glittering with rings.
The couples obliged and were handed four numbered plastic cards engraved with the Chez Clouseau logo, an inscripton with two crossed red wine bottles on a fleur-de-lis background, upon which they were led at a brisk but elegant pace by the maitre d’ to their table.
After dining at a tastefully set table with dishes crisscrossed in an artistically random fashion with sprayed-on green stripes – just missing some small stacks of colorful tidbits – drinking expensive vintage Bordeaux wine, and then paying the exorbitant bill, the couples returned in a financially decimated and bewildered state to the wardrobe to at least recover their coats.
“Our coats please,” commanded Jack, handing the plastic cards to Molly so she could connect the right owner with the right coat.
“Thank you mister,” beamed Molly with a charming smile. Just then a handsome French waiter passed by with a tray holding several full wine glasses, and Molly’s eyes glued onto and followed the garçon, dropping the plastic cards, which fell helter-skelter in a random fashion on the floor behind the wardrobe counter.
“Oops,” she blushed all flustered, “Really, how could this all have happened, y’all.”
She quickly checked the numbers on the plastic cards and handed Jack and Jill and Herman and Lola the coats corresponding to the numbers on the cards and the coat rack.
As the ordered taxi was beeping loudly outside, the two couples grabbed the coats given to them and raced out the door in a rush to leave. Molly didn’t get a tip.
What is the probability that at least one of the four received their own coat on checking this in the taxi? And the probability that one of the couples both received their own coat? That both couples received their respective own coats?
Casino manager Mango is worried about the new dice game “Elevens and Twelves” he invented, as the casino is losing money on it at a rate of about 1%. He fears that casino owner Don Corleone will certainly not be very happy about this.
The game is based on throwing three dice and getting 11 or 12 as a result. The casino pays even money to the player on throws of 11 and the player loses on throws of 12.
Players throw the three dice until they get an 11 or a 12. Then they can make another bet and repeat the process.
Mango had studied probability theory, made some calculations and concluded that the casino would make a profit of about 1% on this game, but, alas, it has not been working out that way.
How did Mango mess this up?