Archive for the ‘Probability’ Category

The Christmas Gifts

Thursday, August 31st, 2017

Fred and Mary were trying to work out how to give five gifts to their three boys for Christmas.

The kids, Henry, Mark and Joe, had told them they wanted to receive gifts more or less at random this year to see how the gifts would be distributed.

“We have five gifts to give to three children. How are we going to do this, Mary? There are so many possibilities it makes my brain spin,” said Fred in exasperation.

“Well, Fred, we have to make sure they all get at least one gift, so that makes it easier than the way you are thinking of, doesn’t it,” Mary said reassuringly.

“You’re right, Mary. The number of ways of giving the gifts so that our kids might not receive even one gift is quite large,” said Fred.

“Do it my way and the matter becomes more simple,” said Mary.

So Mary and Fred distributed the gifts as she suggested.

If they distributed the gifts Fred’s way what would be the probability that one or more of the kids received no gift?

The Die Game

Thursday, August 24th, 2017

Jerry and Slick were sitting at a bar drinking beer and watching a championship football game being televised on the huge wall screen, a bowl of peanuts close at hand.

“Hey Jerry, want to play a game?” said Slick, taking a large sip of his cold beer.

“Sure, what’s it all about,” replied Jerry, munching on some peanuts while eyeing the curvy waitress.

“It’s really simple. I throw this die and if you throw a higher number, you win one point, otherwise I get the point,” explained Slick as he placed a large red die between the beer glasses on the green felt-covered bar counter.

“Let’s try it first, Slick,” said Jerry with a suspicious look.

Slick threw the red die and got a two. Then Jerry threw and got a three.

“Hmm, seems like a good game,” said Jerry with a sly smile.

“Glad you like it, Jerry. How about we play twenty rounds, or until the football game is over – point loser pays the bill?” said Slick.

“Fine by me,” replied Jerry, giving a loud cheer for a goal just made and snatching a handful of peanuts from the bowl.

Slick ordered some fast food and they rolled the die until the end of the football game.

Who do you figure paid the bill, and why?

The Speedy Pigeon

Friday, June 30th, 2017

“Hurry, hurry, try your luck, folks. Get lucky and win a thousand dollars with one shot,” shouted the hawker at full lung power to announce the new game stand at the Jolly Woods Amusement Park.

People flocked to the stand in droves to learn more about this intriguing new game. When they arrived they saw a large semicircular amphitheater with a huge screen about 50 m away facing the spectators. The setup reminded them of a drive-in movie theater of old.

They were told that a yellow virtual pigeon with a blinking red light would be shot out on the screen at a very high speed, leaving a trail of light. To win a thousand dollars a player had to hit the pigeon using a special laser gun each seat was equipped with.

One shot cost $10 and players were told that the chance of hitting the speedy virtual pigeon was 4 in a thousand. Payment was via a card slot at each seat, wins immediately transferred to the card.

Another feature was that for $500 a player could once invite up to two hundred friends to simultaneously take a shot at the pigeon for free.

“C’mon Charlie, lets give it a shot,” shouted one burly onlooker, taking a seat and picking up the laser gun.

How many friends would you invite to this game to have a fifty percent chance of hitting the virtual speedy pigeon and winning five hundred dollars?

The Elevator

Thursday, May 11th, 2017

At the hotel Excelsior, Melvyn had stopped by the security office run by his friend Zack. They were watching the elevator on the split screen monitor, with one screen for each of the ten floors of the hotel and the ground floor.

“Zack, you see the seven people entering the elevator on the ground floor?”

“Sure, Melvyn, especially that hot blonde that just wiggled in before the door closed,” said Zack, chewing on his bubble gum.

“Zack, I’ll bet you a hundred bucks that there will not be a group of exactly three people getting off the elevator at any floor,” said Melvyn.

“You’re on, I could really use a hundred bucks,” chuckled Zack.

“Are you really sure that three persons will be getting off at some floor?” quieried Melvyn, raising an eyebrow.

“That’s what I’m betting on, Melvyn.”

“Well, let’s see how it goes, Zack.” Melvyn made himself comfortable in an armchair to watch the monitor.

What do you figure are the chances that Melvyn will win his bet with Zack?

Can you work out the probability that only one person gets off the elevator at each stop?

The Four Aces

Sunday, April 30th, 2017

Fred was playing poker with his three friends Henry, John and Mack and had just dealt a hand for each.

Fred saw that he had an Ace and wondered what the probability was that each of the other players also had an Ace.

Later in the evening after the game, Fred decided to write a computer program that would deal ten thousand hands of poker to check how often four players each would be dealt an Ace.

Then Fred wondered what would be the similar case for bridge hands dealt, for which purpose he changed a few input parameters for his computer program and received a surprise answer.

On the average, how many of the ten thousand poker hand deals do you think resulted in an Ace for each player?

On the average, how many bridge hand deals did Fred find gave an Ace to each player?

An Olympic Swimming Team

Monday, April 10th, 2017

Jack is part of a ten member Olympic swimming team.

A 500 m relay race is to be run with five contestants.

What is the probability that Jack will be among the five contestants chosen?

The Seating Arrangement

Friday, March 31st, 2017

A dating club arranges meetings at concerts for its members. The club will buy a certain number of random adjacent concert seats for some male and female members. If a man sits next to a woman, the club considers this a date opportunity.

For the coming concert, the club bought 13 adjacent seats for 7 men and 6 women. Some lucky ladies could give a card with their phone number to the man or men sitting next to them.

As an average estimate, about how many cards with phone numbers do you think were given by the ladies at the concert?

A Medieval Tournament

Wednesday, March 29th, 2017

Lord Highcastle of the Moors was holding a tournament in which the first event would be jousting between the three knights Godfrey, Hotspur and the Black Knight. The winner of this event would receive the hand of Lady Marion and a mansion with a large estate.

The knight winning two jousts in a row would get the prize.

Godfrey and Hotspur were to do battle in the first joust, the winner jousting against the Black Knight. If the winner of this match had not already won twice, he would joust against the knight awaiting his turn, and so forth. This to continue until one knight won twice in succession.

Heavy bets were being made in favor of the Black Knight winning within five jousts.

What do you say is the probability that the Black Knight wins within five jousts?

What is the probability that Hotspur rides off with the Lady Marion?

What is the maximum number of jousts needed to arrive at a final winner?

The Blind Mice

Thursday, February 23rd, 2017

“Hi Jack, how are things.” Jill sat down at the café table where Jack was enjoying a cup of coffee and eating a croissant as an afternoon snack to relax after university classes.

“Fine, Jill, but feeling a bit weird,” replied Jack, munching on his croissant.

“Tell me all about it, Jack, I’m listening,” said Jill. She ordered coffee and a donut from the hovering waitress.

“Well, I’m doing an internship for my course on statistics with a Dr. Schnitzelbrenner who is carrying out medical research using blind mice,” said Jack, sipping his coffee.

“And what does Dr. Schnitzelbrenner do with the blind mice,” asked Jill, adjusting the dishes on the table to accommodate the coffee and donut being carelessly dumped by the waitress.

“Well, Jill, he has a large black cat named Schreck – that seems to get fatter by the day – which he uses to frighten four blind mice into running out of their cages in the direction of five entrances, one containing a large chunk of their favorite aromatic cheese,” explained Jack.

“What in the world is Dr. Schnitzelbrenner doing that for,” asked an incredulous Jill. She moved closer to the table and crossed her legs, elbow on a knee holding up her head and gazing attentively at Jack.

“Dr. Schnitzelbrenner says he is testing the orientative capacity of terrorized blind mice compared with blind mice who are not in a state of terror, using a quadruple-blind testing method – all for the purpose of developing a drug for Big Pharma against disorientation,” said Jack nonchalantly as he was ingesting the last piece of croissant.

“How is this a quadruple-blind test,” inquired Jill.

“Well, Dr. Schnitzelbrenner is using four blind mice,” explained Jack.

“Oh, I get it, that’s why it’s a quadruple blind test,” laughed Jill.

“And how will Dr. Schnitzelbrenner determine whether his medicine works or not,” asked Jill.

“Well, if all the mice go into different entrances in any one trial, a light switches on and a buzzer sounds. It is my job as a statistician to record these trials, with or without signals going off,” explained Jack.

“According to Dr. Schnitzelbrenner, if the terrorized mice all head for the entrance with cheese, then they are not to be considered disoriented and the drug works. If they all go to different entrances, then they are randomly well disoriented,” continued Jack.

“Hmm, tell me more,” said Jill.

“If not too many mice die from the active substance in the drug, Dr. Schnitzelbrenner hopes to win the Nobel Prize for this new medicine, which he says will aid senior citizens who suffer from disorienting illnesses,” added Jack.

“Very quaint experiment, I must say, but senior citizens will not normally be in a state of terror,” observed Jill.

“According to Dr. Schnitzelbrenner, senior citizens actually do suffer from a suppressed subconscious fear-of-death-inspired terror, and that is really why they easily become disoriented,” explained Jack.

“I see. How long will you be working with Dr. Schnitzelbrenner, Jack?”

“Just another week or so, after I calculate the probability of randomly well-disoriented blind mice each winding up in separate entrances, which I find a bit difficult,” confessed Jack.

“How about a walk in the park, Jack, to get your mind off Dr. Schnitzelbrenner’s terrorizing research, and I’ll give you some hot tips,” she suggested, emptying her cup of coffee.

“Capital idea, Jill.”

They paid and left the café.

Can you work out the probability of a group of four blind mice all going into a separate entrance in any single trial with five entrances?

Spies at Emperor Wang Shu’s Court

Saturday, December 31st, 2016

One summer evening during the hour of the horse, the Chinese emperor Wang Shu summoned his spymaster Lang Dang to his private suite in the palace. He was worried, as he had received information from the palace cook Chu Shi that his enemy, the warlord Dui Yuan, had planted ten spies among his royal guards.

“Lang Dang, there are ten spies among my hundred royal guards. They all have a dragon tattoo on the sole of their right foot. You are to find at least one before the hour of the rat,” commanded Wang Shu.

“Yes, heavenly ruler, this is already done,” said Lang Dang with a deep bow, his thin grey beard reaching large black sleeves into which he had stuck his hands.

“You will interrogate Dui Yuan’s spies you discover so as to reveal the identities of the others. You should know that the leader of the spy ring has a dragon tattoo on the soles of both feet,” added emperor Wang Shu. “Find him and Mei Ling is yours as a concubine,” he offered.

“Ah so.” Lang Dang retreated bowing deeply and shuffled off eagerly, making some mental calculations.

To avoid arousing any alarm and achieve rapid results, Lang Dang decided to round up ten royal guards at random and check their feet. That should do the trick. He would interrogate them until their leader was revealed.

What is the probability of Lang Dang finding one or more of Dui Yuan’s spies on selecting ten royal guards for foot inspection?

How many royal guards must Lang Dang select to be 90% sure to find a spy?

What is the probability of Lang Dang finding the leader of Dui Yuan’s spy ring in a sample of ten royal guards?