Ken's blog

“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?

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?

“Langhorne, we have to reduce the number of stiffs from gunfights, Boot Hill is getting full,” said Sheriff Wilbur Harp of Gravestone, Arizona to sheriff Langhorne Dillon of Dead Man’s Butte.

“Some tighter rules are needed for sure, Wilbur,” agreed Langhorne Dillon, emitting a brown streak of chewing tobacco toward a spittoon in the corner of the Bird Cage saloon.

“I propose to limit gunfights to between 12 noon and 1 pm, while people are eating,” said Wilbur Harp, sending his own brown streak to the spittoon over a cocktail waitress dressed in red who was leaning over to refill their whisky glasses.

“At your service, boys,” she smiled and wiggled away to the bar.

“Yes, Wilbur, and they can show up at any time between noon and 1 pm and must wait for fifteen minutes. If the other party doesn’t show up, the gunfight is cancelled, violations punished with jail time,” added Langhorne Dillon.

“I agree, Langhorne, but for Gravestone I prefer a ten minute wait,” interjected Wilbur Harp.

“Ok, Wilbur, and the procedure should be monitored by our deputies,” added Langhorne Dillon, stuffing his nose with some snuff.

“To limit things further, they will only have one bullet in the barrel, can spin the barrel and shoot up to three times,” said Wilbur Harp, spinning the barrel of his Colt .45 single-action revolver, causing a nervous ducking reaction of people at the tables in the saloon.

“That’s ok, Wilbur, but in Dead Man’s Butte I prefer that they don’t spin the barrel, but keep shooting up to three times – to be monitored by my deputy Leonard,” said Langhorne Dillon.

“Fine by me, Langhorne,” said Wilbur Harp, sending another brown streak through the air in the direction of the spittoon, just missing the bartender walking by. “Let’s compare statistics at the end of the year to see which procedure is the most effective for reducing the number of stiffs, yours or mine,” he added.

“Right, Wilbur, assuming our current rate of one gunfight per day continues for a year, I’ll ask my deputy Leonard to work out the figures,” said Langhorne Dillon. “With the Clinton gang in operation rustling cattle from Mexico, I don’t figure the current rate will drop much,” added sheriff Dillon.

“Yeah, I know what you mean, Langhorne. Settled then, let’s see if Leonard’s calculations will be accurate,” sheriff Wilbur Harp, emitted another brown streak and downed a shot glass of whisky.

Given this information, can you work out how many coffins  the undertakers of Gravestone and Dead Man’s Butte are likely to need, respectively, during the coming year?

“Mommy, I want my cereal, my Munchy Crunchies,” pleaded Junior at the breakfast table. Mommy placed the cereal box, almond milk and honey in front of Junior.

While munching away Junior was avidly reading the promotional text on the back of the cereal box.

“Mommy, it says here that if I collect five famous Disney character cards, I will get a prize – a scout compass. There is one card in each box. Let’s quickly buy five boxes of Munchy Crunchies so I can get the compass for my Boy Scout trips,” suggested Junior with a bright enthusiastic look.

Daddy overheard Junior. “It doesn’t quite work that way, son. Maybe the five boxes we buy all have the same character card, or maybe just one different card.”

Junior was not discouraged. “It also says that cards have one of five colors on the other side: red, blue, green, yellow and orange,” informed Junior. “If all of the five character cards have a red back side, I can also win a Speedo bicycle. Please, daddy, we have to buy more boxes today,” pleaded Junior, munching away faster.

“Hmm,” said daddy.

“And, daddy, it also says that in one of every thousand boxes there is a gift card for a free trip to Disneyland for a family of three, with all expenses paid,” said Junior beaming.

Daddy became more interested. “Hmm, I will work out our chances for all this,” he said prudently, grabbed his calculator and went to his work room.

“Thanks, dad.” Junior was beaming with pleasure, thinking of the great prizes he would win and how much he would enjoy going to Disneyland.

Mommy cleaned the table. Then she made two cups of coffee and went to talk with daddy.

Can you help daddy work out how many cereal boxes on the average he will have to buy for Junior to obtain the five Disney characters and win a scout compass?

How many cereal boxes on average will win the red bicycle?

And how many boxes on average for a family trip to Disneyland?

Jackson was concentrating on the three cups filled with dice placed in front of him on the table covered with green felt, wondering which to choose.

“Some say that the result of any of the cups will be the same, just a fifty percent chance,” said Snipes the slick gambler. “Which of the cups do you say has the best chance?”

Jackson had bet Snipes a thousand dollars that he could pick the right cup and prove his point with a hundred throws of each cup.

Respectively, the cups – colored red, green and blue – contained six, twelve and eighteen dice. The issue was regarding the event probability of a throw of the cup with six dice giving at least one six, of the cup with twelve dice giving at least two sixes, and of the cup with eighteen dice giving at least three sixes. The question was which cup had the greatest event probability.

“Well, what do you say, Jackson,” said Snipes with a sly grin, a golden tooth glinting. “Which cup do you choose, the red, green or blue one?”

“Hard to say,” mumbled Jackson, thoughtfully stroking his bearded chin.

“Could be close, indeed. Snipes, would you mind if I check my runes?” Jackson pulled out a decorative brown leather bag containing rune stones and placed it on the table.

“No problem, Jackson. Take your time,” grinned Snipes, whose mental equipment did not believe in mumbo-jumbo.

Jackson rummaged in the rune bag with a pensive expression on his face and picked out three rune stones, which he carefully placed in a row on the green felt.

After a quick look at the runes he said, “Snipes, I’ll take that one,” indicating a certain cup on the table.

“Well, you picked the right one, Jackson. Just luck, I guess,” said Snipes with a mournful expression on his face and handed over the thousand dollars to Jackson.

“What about a game of Black Jack, eh?” Snipes placed a deck of cards on the table with a swift movement of his hand.

Jackson replaced his runes and walked away.

Which would you pick, the red, green or blue cup? And why?

“Secundus, send seven scouts to gather intelligence about the positions of the barbarians!” barked generalis Clavdius Maximus to his centurion Celerem Secundus.

“Ave generalis, iam facta est.” Celerem Secundus ran off to dispatch seven scouts into the thick woods.

…These barbarians are primitive rabble. These tribus Germanicus will never develop a civilization as technically advanced as ours.

Clavdius Maximus wondered how many scouts would reappear. Lately only six out of ten scouts had found their way back to report what they had seen. The barbarians were very clever at spotting Roman scouts and would dispatch them quickly.

…Would at least five of the seven return?

What do you figure is the probability of at least five of the seven scouts reporting back?

Pierre, the First Mate, was sitting in the mess room of the Nautilus flipping cards from a Tarot deck. He would periodically stop and write some numbers in a small notebook near the cards.

“What are you doing, First Mate?” said Captain Nemo, who had stopped to look as he was walking by on his way to the bridge.

“Sir, I am checking to see how many cards on the average I need to draw from a shuffled Tarot pack before I get an Ace,” said Pierre with a bright smile as he drew The Hanged Man.

“Why don’t you just calculate it? It’s much faster,” grinned Captain Nemo.

“I thought about doing that, but there are so many combinations on how to get an Ace,” moaned Pierre scratching his head as he pulled The Fool.

“Why don’t you think about how not to get an Ace instead,” suggested Captain Nemo, continuing on his way to the bridge.

“Hmm, that’s not a bad idea, Sir,” the First Mate replied and started to write some symbols in his notebook.

What do you figure is the average number of Tarot cards that need to be drawn from a well-shuffled Tarot pack before an Ace appears?

The Wild Weed Band had just completed its concert on a fine day in August and was reaping enthusiastic and noisy applause from the five thousand fans extending far beyond the band stage.

“As it’s my birthday today, I feel generous,” shouted the band’s singer Magno Solo jubilantly.

“All of you who have your birthday today are invited to come up here with your concert ticket and receive a 5,000 dollar check as a token of my great appreciation of my fans,” Magno Solo held his glittering arms out in a welcoming gesture.

Rapidly a group of people with smiling anticipative faces materialized on the stage.

Bongo King, the drummer, stood ready with a bundle of checks, handing them out one by one to the overjoyed people in the group. He gave each one a big hug and a flower.

“For an additional special prize of 20,000 dollars, give your ticket to our guitarist Marco, who will convert your ticket number to a day of the month number with his little calculator. You’ll win if your number converts to today’s date,” exclaimed Magno Solo with a regal bow.

The group on the stage rushed to Marco, who checked each ticket and converted the ticket number to a day of the month number, announcing any winner with a fanfare from the band.

The concert ended with enraptured winners loudly singing the song “For he’s a jolly good fellow,“ accompanied by the Wild Weed Band to appreciative shouts and whistles from the audience.

About how many persons had the same birthday as Magno Solo?

How many persons do you think won the final prize of 20,000 dollars?

It was a sunny Caribbean day with fluffy clouds in a blue sky. A pirate ship was smoothly gliding along on a mild breeze driving its white sails.

The ship was Captain Hook’s and Jacko was walking the plank for insubordination. Jacko was blindfolded and had already drunk a jug of grog given him to send him happily on his way to the wet kingdom of Neptune. Jacko was standing on the very edge of the long plank, about to plunge into the waves.

Peg-leg, the bosun, was busy taking bets from the crew while his blue-winged, yellow-bellied parrot screeched “Take your bets, take your bets…” at full volume from his shoulder.

From previous experience, Peg-leg, an amateur mathematician, had worked out that after downing this amount of grog plank walkers would at any given moment step forward with a probability of 33%, or step backward with a probability of 66%. The length of the step was somehow constant. All this seemed to be induced by the grog mixture.

The basis for betting had been established by Captain Hook, who, being a humane type, had decreed that plank walkers who managed to step their way back onto the ship after drinking a jug of grog would be pardoned.

So, Peg-leg was busy taking bets from the crew as to whether Jacko would make it back on board the ship, or plunge into the laps of mermaids.

Jacko was about to take his next step.

What do you figure is the probability that Jacko joins the mermaids?

In ye legendary olden times in Sherwood Forest, Robin Hood, Little John and Friar Tuck set out to bag a great deer for a large celebration to be held on the occasion of King Richard’s return to England from the Crusades.

Robin’s probability of bagging a deer with his bow and arrow is 0.9, Little John’s and Friar Tuck’s 0.7 and 0.5, respectively.  The feathers of their arrows are green, blue and red, respectively. They will fire simultaneously.

Shortly they track down and bag a large deer and haul it back to the camp on a wagon to be received by a multitude of cheering band members preparing to make merry.

What is the probability of the cook and his helpers finding a green, blue and red arrow in the deer?
And the probability of only a red arrow?
What about just a green and a blue arrow?
Or just a green arrow?
Or at least one arrow?