The Dice Club Project

September 30th, 2018

One Saturday afternoon, as Jack, a new member, entered the Snake Eyes Dice Club he saw many people busy throwing dice at various felt-covered tables.

“Why are you repeatedly throwing that die, Vince, won’t you get a sore arm,” asked Jack.

“Hope not. I’m checking how many throws are needed to get all die faces from one to six to appear at least once,” said Vince.

“Any conclusion so far?” asked Jack.

“Seems that about 14 throws will do the job,” said Vince, after checking some marks on a notepad. “We’re running a project to check the results for different numbers of dice,” added Vince.

“So you’ll soon be using two dice to see how many throws are needed to get all the doubles?” said Jack.

“That’s the next step,” agreed Vince, “probably take quite a bit longer. Maybe I’ll pass the job on to Joe over there,” said Vince, rubbing his elbow.

“Charlie at the big table yonder is working on getting all the triples, but he’s been at it for a really long time,” said Vince.

“Anybody working out the probability for n dice?” said Jack.

“Yeah, my cousin Lennie at the desk over there is doing the theory and checking it out on a PC. He’s quite good at math,” said Vince.

“Good luck with the project, Vince. Seems a bit complicated to me, so I’ll be heading for the lounge,” said Jack.

Can you offer a formula for calculating the number of throws with n dice needed to get all the n-tuples?

 

Bags with Gold and Silver Coins

September 29th, 2018

“Jack and Jill, which bag do you choose for a chance of winning ten thousand dollars,” hailed the cheerful game show host Dusty Farlone, “the blue bag with three gold and seven silver coins, or the red bag with four gold and three silver coins. What do you say?”

“Dusty, what were the conditions, again?” asked Jack.

“You can pick six coins out of the blue bag with ten coins, or you can pick six coins out of the red bag with seven coins, in both cases replacing the coin each time. Whichever bag you choose, if you manage to pick three gold coins, you will win ten thousand dollars plus all the coins in the bag, of course,” said Dusty Farlone, smoothing his slick hairdo.

“Our charming Marlene will bring you the bag you select so you can pick the six coins, and help you put them back into the bag again,” said Dusty Fallon, indicating the svelte Marlene. “You have a minute to decide while the music plays.”

Jack turned to Jill. “You took a course in probability theory. Which bag seems the most promising for winning the prize, Jill, the blue or the red one?”

“Hmm… good question,” said Jill, “quick, hand me your scientific calculator, Jack.”

Which bag would you say has the greatest probability of success for picking three gold coins and winning the game show prize of ten thousand dollars?

A Basket of Fruit

September 24th, 2018

“I hear the jingle of the fruit vendor Zoltan passing by, Lennie,” said mommy. “Won’t you please run out and buy some fruit for your lunch box next week, and take this basket.”

“Sure, mommy.” Lennie ran out with the basked to catch Zoltan before he rode away on his bicycle stand.

“Mommy wants me to buy some fruits for my lunch box,” said Lennie eagerly as he arrived breathing heavily at the fruit stand.

“Fruits are good for you, young man,” replied Zoltan, “I have five apples, four pears and three plums. What would you like for your lunch box.”

“I’ll take three apples, two pears and one plum, please,” said Lennie.

“That will be one dollar and twenty cents, my little friend,” said Zoltan.

“Ooops, I forgot to bring some money,” said a flustered Lennie.

“Young man, if you can tell me how many different baskets you could fill with apples, pears and plums, taking none or any number of each of these fruit types each time, you won’t have to pay,” laughed Zoltan, who had taken a shine to the bright boy.

“Ok, sir, I’ll think about it,” said Lennie, scratching his head full of hair…

“Mommy, here are some fruits for my lunch box next week,” said Lennie, handing over the basket with the fruits.

“How much did it cost,” asked mommy.

“I didn’t have to pay because I gave the right answer,” said Lennie.

Can you work out how many baskets Lennie told Zoltan could be filled with the fruits on display?

 

The Small Bouquet

September 20th, 2018

One cold, snowy winter evening on Valentine’s Day, Jasper was in a hurry to buy a bouquet for his sweetheart Molly while rushing on the way to her birthday party.

The only flower shop Jasper found open among the swirling snow flurries was the Thrifty Flower Emporium, where a total of fifteen flowers were left, namely five Lily of the Incas, four anemones, three carnations and three daffodils.

Since Jasper didn’t have much money on him, he decided to buy just three flowers, one for each of the years he had known Molly.

However, there was the problem of choosing the right combination of flowers, which caused a mental block to occur in Jasper, and he stepped from one foot to the other in complete indecision.

The shop attendant, Susan, noticed Jasper’s panicky bewilderment, and informed him that they had just bought a Smart Pick flower randomizer which could select from among the available flowers and have Jasper’s bouquet for Molly ready in a jiffy.

This good news put a big smile on Jasper’s face, and in no time Jasper walked out of the Thrifty Flower Emporium proudly exhibiting a bouquet with three flowers in his hand, in a sprint to reach Molly’s birthday party on time.

What would you say were the number of possible triple-flower bouquets that could be made with the fifteen flowers available at the Thrifty Flower Emporium?

And what is the probability that all the flowers were different?

 

The Chinese Round Table

August 31st, 2018

One fine evening, Jack and Jill were dining at the Fu Manchu Magic Noodle House, their favorite Chinese restaurant. They sat at a round table served by their regular waitress, Wang Shu.

A number of dishes containing delicious tidbits for plucking off with their chopsticks were situated in front of them on a rotatable round tray.

Jack and Jill sampled the many delectable Chinese morsels to their heart’s content.

“Pick a fortune cookie, Jack,” said Jill, pointing at the decorated cookie jar in the center of the table.

“The cookie says: ‘Good fortune. Soon get special dessert’,” said Jack.

“Hmm,” said Jill, “what does it mean, Wang Shu?”

“Now, special desert,” announced Wang Shu while clearing up after dinner.

Wang Shu placed four dishes containing fried ice cream balls and spicy chocolate truffle balls – delicious Chinese desserts.

In each of the first three dishes there were two fried ice cream balls and eight spicy chocolate truffle balls. The fourth dish contained six fried ice cream balls and four spicy chocolate truffle balls.

“Fortune cookie offer special dessert,” said Wang Shu.

“How generous,” replied Jack.

“Bring it on, Wang Shu,” said Jill.

“Close eyes,” said Wang Shu, “you too,” addressing Jill. Wang Shu spun the dessert tray with the four dishes.

“Please, keep eyes closed and pick one dessert from dish,” said Wang Shu to Jack. Jack picked a dessert ball, which Wang Shu quickly replaced from a large pot containing both dessert types.

Jack opened his eyes and saw a fried ice cream ball melting in his hand, which he promptly ate to his great satisfaction.

“You guess which dish ball from, you no pay. Boss say special promotion,” informed Wang Shu.

“We like that,” said Jill.

Jack scratched his head and then wrote some figures on a napkin, Jill peeping on. Then Jack pointed to a dish.

“Correct dish you choose. Boss say, this time free, next time you pay plenty,” said Wang Shu with a big smile.

Jill was already busy munching away at the balls on the dessert dishes.

“Doggie bag, please,” said Jack.

Was Jack just lucky or had he worked out a probable dish to pick?

Which of the dishes do you think Jack chose?

The Flying Card

August 28th, 2018

At the Lucky Duck casino Nick, the dealer, was sitting at a green felt table and practicing fancy acrobatic shuffles with a deck of cards when suddenly one card flew away and sailed into a nearby plant bed with some petunias and foliage.

“I’ll bet ya 100 bucks you can’t figure out if da card dat just flew away from your deck is a red or a black one,” said Carmine, the floor overseer who happened to be walking by and observe the event.

“You’re on,” said Nick, “on one condition.”

“What’s dat,” replied Carmine with some suspicion.

“That I draw 13 cards from the deck at random,” said Nick.

“Ok, sure,” said Carmine, “no problem, Nick, go ahead an pull dem.”

Nick then drew 13 cards at random from the deck and they all turned out to be black cards.

Then Nick pulled out a calculator, a notepad and a pen and got busy making some calculations.

After some minutes had passed Nick proudly announced: “The card is red, Carmine, go and check it out.”

Carmine walked over to the flower bed and picked up a red card.

“Jeez, Nick, how did ya figger dat out?”

“Give me the 100 bucks and buy me a beer and I’ll tell you all about how to make a pretty sure bet,” said Nick.

 

Can you figure out how Nick knew that the card most likely was a red one?

The Faulty Die

August 24th, 2018

Jack was sitting at a café table reading a magazine when Jill sat down and ordered some pastries and a hot chocolate.

“What are you reading with such a puzzled look on your face,” said Jill, munching a delicious apple strudel with a flaky crust.

“It’s a puzzle about a guy who is given a bag with 100 dice and told that one of the dice is faulty since it has six dots marked on all of its faces,” said Jack. “Then he is told to reach into the bag and pick one die.”

“So what’s the puzzle all about, then?” asked Jill, taking a small sip of her hot chocolate.

“You’re supposed to work out how many sixes in a row the guy has to throw before he can be 90% sure that the die he is throwing is the faulty one,” said Jack.

“Hmm… since there are 100 dice, it seems to me that you’d have to throw an awful lot of sixes to be sure the die was faulty,” said Jill.

“Maybe,” said Jack, “but we have to give mathematical proof of the solution to the puzzle.”

“What’s the prize?” said Jill finishing off her apple strudel and taking a large sip of her not so hot chocolate.

“Jill, it’s ten thousand dollars for the correct answer,” said Jack.

“What are we waiting for, Jack!! Why don’t you get busy on figuring out the solution,” said Jill enthusiastically, “then we can rent an apartment in Mambo Bay for a great vacation.”

“Hmm…,” said Jack, “’Bay’ gives me an idea.”

Straight vs. Full House

August 14th, 2018

On another fine evening, there was a heated discussion at the Royal Flush Card Club as to how many occurrences of Full House and Straight would occur in the course of 30 throws of five poker dice.

Bing Jones III was betting $100 against Franz Fingerflitzer that he would get at least two instances of a Full House, the latter claiming at least two Straights during the trial – ties leading to a rematch until a winner emerged.

Lots of paper was expended and the whisky flowed freely while members of the Royal Flush Card Club busied themselves calculating the probabilities involved for throwing a Full House and a Straight.

However, accuracy and interest faded rapidly as whisky was poured, and this task was abandoned in favor of registering bets and proposing exuberant toasts, expectations rising to a high pitch, as if prior to a championship horse race.

Who do you think won the bet, Bing Jones III or Franz Fingerflitzer?

The Same Sun Sign

July 31st, 2018

“Jack, I’ll be taking a course in Astrology soon,” said Jill.

“That’s nice,” said Jack, “How many people will be on the course?”

“Twenty-five or so, I was told,” said Jill.

“How many people do you think will be Geminis, like you?” said Jack.

“No, idea. What would you say, Jack?”

Jack told her and it turned out he was pretty right.

What would you say was the number of other Geminis Jill met on the Astrology course?

 

The Deck Split

July 29th, 2018

At the Royal Flush card club one evening, there was a lively discussion among its members about splitting a deck of cards into two equal piles after a thorough shuffle, in which, respectively, there would be exactly ten red cards in one pile and exactly ten black cards in the other pile.

Bets were taken for twelve trials to check whether this event would occur at least once.

How much would you bet on this event occurring in the course of twelve such splits?