May 30 – It’s A Snap

One of the amazing things about science is that what you learn when you study one thing often applies to something else that you wouldn’t think was related at all. In today’s adventure, Daniel, Peter, and Mary discover that spaghetti, earthquakes, animals, and authors all have a secret connection.

 

When Peter’s mother walked into her kitchen, the last thing she expected to see was pieces of spaghetti lying all over the table with Mary and Peter laughing over them. But that’s exactly what she saw. Peter and Mary were sitting on opposite sides of the table, watching each others’ hands very closely as they took turns picking up pieces of spaghetti and snapping them.

“Hey,” Peter’s mother said. “What are you two doing?”

“We’re doing an experiment,” Peter explained. “I read somewhere that spaghetti always breaks into two pieces when you snap it and Mary didn’t believe me. So we decided to see if it was true.”

Looking at the pile of broken spaghetti, his mother shook his head.

“I know that you are supposed to replicate your results, but why so many times? ” she asked. “You should have been able to answer it with the first few pieces.”

“Well,” Peter said. “I can get the spaghetti to break into two pieces pretty consistently but Mary almost always gets three pieces except when she gets four. We’re trying to figure out why.”

“Well, it certainly looks as if you’ve done a pretty thorough job. But have you even decided what the variables are in your experiment?”

“Well, no, ” the two admitted sheepishly.

“So what you are really doing is making a mess,” his mother admonished. “If you don’t know what you’re doing, then you can hardly know if you are doing it right. Tell you what. If you’ll sweep all of this mess into a bowl so we can use it for dinner tonight -”

“Oh, boy! Spaghetti!” Mary interrupted.

“-then I’ll show you a real experiment with spaghetti. Deal?”

By way of agreeing, Peter and Mary quickly jumped up then gathered up the spaghetti pieces into a bowl. Once the table was clear, Peter’s mother put five pieces of whole spaghetti in front of each of them along with a penny.

“Here’s the experiment,” she explained. “What you’ll do is pick up each piece of spaghetti in turn, working from left to right. The other person will flip their coin. If they get heads, then you have to snap the spaghetti into two pieces of equal length by holding it in the middle; you then put the two pieces at the far right side. If the other person gets tails, then you put the piece of spaghetti at the far right and you don’t snap it. Got it?” At their nods, he continued. “You’ll each flip your coin forty times. So you know the variable – the coin flips. Here’s the question: How many pieces of spaghetti will you have at the end?”

“Well, we start with five pieces,” Peter said. “So each one will get eight flips. If they get broken half the time, then each piece should get broken four times giving us two pieces each time and we’ll have five times four times two or forty pieces.”

“I don’t think that’s how it will work,” Mary said. “Remember that this is random. So some of the pieces will get broken more than once and others won’t get broken at all. So it will probably be less than that; I think it will be closer to thirty pieces.”

“Good,” Peter’s mother said. “And notice that you both added an extra prediction. Peter seems to think that the pieces will all be about the same size but Mary, you think that they’ll be different sizes.”

“That’s right,” Peter said.

“Well, there’s only one way to find out,” Peter’s mother said. “Start flipping!”

What do you think will happen? Do the experiment!

 

 

 

 

 

 

For the next ten minutes, the kitchen was filled with the sound of flipping coins and breaking spaghetti. Soon, some of the strands looked less like spaghetti and more like toothpicks while other strands stayed long. At the end of all the flipping, they quickly counted up their pieces of spaghetti.

“Hey! I only got twenty-five pieces!” Peter exclaimed.

“Yes, and one of yours is still whole,” Mary added. “I have thirty pieces but none of mine are longer than a half.”

“So what gives?” Peter asked. “I kept track – we both got the same number of heads.”

“Ah, but did you get them in the same order?” Peter’s mother asked. “Because you kept cycling through the pieces in a strict order, getting heads or tails at a different time meant that you’d get a different length. But, if it makes you feel any better, if you had done more flips on more strands of spaghetti, then you would both have gotten almost identical results. Yours are actually pretty close, given how few times you flipped the coins.”

“How many would we have needed in order to get the same result?” Mary asked.

“An infinite number,” she replied. “But if you’d flipped your coins about 1000 times and used ten strands of spaghetti each, then you two would have been much, much closer.”

“But why didn’t we get forty pieces each?” Peter asked.

“It is just like Mary said – you forgot that you’d be flipping on individual pieces and not the whole strand each time; there’s a fun little math equation that describes what will happen. But what’s really cool about this is that you can do the math backwards and find missing things. Take asteroids for example,” she explained. “We know how many we have found and what their approximate sizes are. By applying that equation, we can predict how many we haven’t found.”

“Cool! Is that why they say that we know where 90% of the big ones are?” Mary asked.

“Yes. But the math doesn’t just apply to rocks in space; it also tells us what the distribution of rocks in an avalanche will look like, and how the number of earthquakes will change with size, and even the number of fish of a given size that are hiding in a lake. Alfred Lotka even used it to predict the number of scientific papers that would be written in any given year.”

“What? How did he do that?” Peter asked.

“He just looked at the number of papers that were written by an scientist who had already published a paper. Let’s say that in one year, there were 500 scientists who each published a paper and that the next year just 125 of them published another paper. Then he knew that in ten years there would only be five of them who wrote a new paper. If you then add up all of the people who were doing research and assumed that they stopped after ten years, you’d just add up the number of scientists publishing for the first time, the second time, and so forth, and discover that there would be 774 papers published that year,” Peter’s mother said. “The neat thing is that it isn’t just papers that you can do that way; it also tells you how emails would be written in a given year and how many blog posts or how many television shows would be filmed.”

“That is so cool!” Mary said. “I didn’t know that one equation could do so much.”

“Well, there is one thing that it can’t do,” Peter’s mother said. “It can’t clear the table so I can make spaghetti!”

With a quick grin, the two hungry young scientists turned to the table and started cleaning up their experiment.

 

 

About the scientist

In 1918, an influenza pandemic swept across the globe and killed one out of every twenty people alive at the time. It was the greatest health disaster since the Black Death and inspired many scientists to find new and better ways of predicting disease outbreaks. One of these was Alfred Lotka who was trained as both a biologist and a mathematician. He had been working on ways of predicting malaria outbreaks and soon discovered that his work was more widely applicable than he had expected.

He started looking at the number of papers published by scientists, At first, he was just curious about why so many scientists seemed to write fewer papers as they got older but he was surprised when he found that he could actually predict how many papers would be published using a simple formula. His result turned out to be a special case of a more general equation that governs everything from animal populations to gravel size to the number of sunspots.

 

March 14 – Completely Normal

Are scientists normal? In today’s Secret Science Society adventure, Peter and Mary discover what the word means to scientists and why math can be important!

It was another typical Monday afternoon. Peter and Mary had finished lunch and were heading toward their favorite class. They didn’t like science class just because they wanted to be scientists; they also liked the way that their teacher, Mr. Medes, always managed to give them an interesting experiment to do. As a result, they frequently tried to predict what the experiment would be while they ate lunch. Today’s discussion had spilled over and continued as they walked to class.

“Well, it is Monday, so we know that it will be about math,” Mary said.

“Sure, but that doesn’t narrow it down much. Remember last week, when he had us coloring maps all class period?” Peter replied.

“And the week before, when we played with clay,” Mary said as they came to the classroom door. “Well, there’s only one way to find out – let’s go ask Mr. Medes!”

Mr. Medes was waiting for the class as they filed in and found their seats. As soon as all thirty-two students had come in, he started in on his lesson for the day.

“Welcome, welcome! Today is Monday, and that means math! For today’s lesson, we’re going to follow in the footsteps of that great gambler, Pascal. We are going to flip coins!”

“What does gambling have to do with math?” Peter asked.

“Believe it or not, most of the early work in statistics was done by Pascal and other gamblers as they sought a way to make more money. Pascal even invented a gambling machine that we know as the roulette wheel; he did it to see if he could create a perpetual motion machine. Instead, it turned into a perpetual money machine. But we won’t need anything that fancy today. We just need pennies.” As he said that, Mr. Medes passed out a penny to each of the students.

“Now, this is a very simple experiment. What we are going to do is flip the coins. If they are ‘fair coins’, which means if they are evenly balanced, then about half of them should come up heads and half should come up tails. Ready? Everybody flip!”

Suddenly the air was full of flipping coins. When the coins had stopped flipping, Mr. Medes asked for a show of hands from those who had heads.

“OK, we had 17 heads and 15 tails,” he said. “Let’s try that again, just to make sure that wasn’t a fluke.”

They repeated the test four more times, getting about the same distribution of heads and tails each time. Mr. Medes wrote 32 on the board then wrote the last result (14 heads and 18 tails) below it before turning back to the class.

“Good!” Mr. Medes said. “That proves that these are fair coins. We got about half heads and half tails each time, and everyone got a different set of heads and tails. But that raises an interesting question. What do you think would happen if we flipped the coins twice and kept track? How many times would we get either two heads or two tails in a row?”

“Well, you’ve got a fifty-fifty chance of getting heads after a tails, so you’d get it about half the time,” Mary said. “There are thirty two of us, so there’d be sixteen people with two heads or two tails.”

“No, you’re wrong,” Peter said. “You’ve only got a fifty-fifty chance of getting heads or tails on the first flip, so there should only be half as many. We’d have eight people with two heads or two tails. Everyone else will have a head and a tail or a tail and a head.”

“Well, there’s only one way to solve this problem,” Mr. Medes said. “Everyone take out a piece of paper and pencil to record your observations. Flip the coin and write down if it was heads or tails, then do it again.”

What do you think will happen? Do the experiment!

 

 

 

 

For a moment, the classroom was silent as everyone flipped their coins and wrote down the results. As soon as the last penny was put down, Mr. Medes spoke up again.

“OK, how many of you got two heads?” Nine hands went up, and Mr. Medes wrote that on the board. “How many got two tails?” Seven hands went up; Mr. Medes once again recorded the number, then put ‘16’ between them saying “And that leaves 16 people with either a head and a tail or a tail and a head.”

“Where did I go wrong?” Peter asked. “There should have been half that many!”

“Actually, you fell into the same trap that frustrated many of the mathematicians who discovered these rules. They forgot that everyone gets either a head or a tail for their first flip and that the real question is how many people will get a second head or tail. We can write it in math this way:”

Turning to the board, he drew

Second flip

Heads (F) Tails (T)
First Heads (F) FF FT
Tails (T) TF TT

“Hey! That looks like the truth tables my mom uses when she programs!” Peter said.

“That’s right; this branch of math lies at the intersection of programming and statistics. It is also useful in genetics and many other fields. If you count it up, you’ll see that there are two boxes where you get a mixed result and two where you get a ‘pure’ result. So, if you had written it down like this, you’d have known the right answer without needing to do anything more. So – how can we decide how many ‘pure’ results we’ll get if we flip the coins three times?”

“We make another truth table!” Peter said.

“Right,” Mr. Medes replied as he drew a table on the board.

Next flip

Heads (F) Tails (T)
Earlier flips FF FFF FFT
FT FTF FTT
TF TFF TFT
TT TTF TTT

“We’ve got eight possible outcomes, and two of those are ‘pure’. So we should get one eighth of thirty two, or four, people with three heads in a row and the same number of people with three tails in a row; that means we should have eight ‘pure’ results. Now let’s flip the coins and see what happens. After you’ve flipped, raise your hand is you have three in a row of the same, heads or tails.”

Everyone again flipped their coins. The class looked around and burst into smiles as seven people raised their hands.

“Why didn’t we get exactly eight?” Mary asked, with a puzzled look.

“Because this is statistics, which works best for a large number of flips. If we had 320 people flipping the coins, we’d probably get somewhere between 75 and 85 pure results. If we had 3200 people flipping coins, we’d get somewhere between 780 and 820 pure results. As you flip the coin more often, random errors even themselves out and you come closer to the true value. Let’s try an experiment to see why. Mary, if you flip a coin, what are the odds of it coming up heads?”

“It will be heads half the time,” she replied.

“Right. Now Peter, flip your coin and tell me what it shows.”

Peter quickly flipped his coin. “It came up tails!”

“Now that doesn’t mean that Mary was wrong. What it means is that you don’t have enough flips, what a statistician calls ‘a significant population’, to decide what the odds are. The only way to tell the odds is to flip a coin a lot of times. And the more times you flip it, the bigger the population and the less chance that some error has crept into your results. As a matter of fact, a mathematician named John Kerrich tossed a coin 10,000 times while he was being held as a prisoner of war by the Nazis. He got heads 5,067 times and tails 4,933 times. If he had only flipped it once, he would have only gotten a head or a tail, just like Peter did. So bigger is better in statistics.”

At that point, the bell rang and Mr. Medes said “Don’t flip out – I’ll see you tomorrow for more science!”

March 7 – Seven Impossible Things

Ah, Saturdays! Is any day better? You get to sleep late, you get to watch cartoons, and (best of all) you get another Secret Science Society adventure! Today, Mary and Peter discover that there are somethings that man is not meant to know…

Most days, Peter and Mary got along well together. They both liked the same things, and they both wanted to be scientists. But every once in a while, they would fight. And, as is often the case with friends, when they did fight it was usually about something stupid. Today was no exception.

“There is so!” Mary shouted.

“No there isn’t!” Peter insisted. “There isn’t anything that is impossible!”

Attracted by the noise, Peter’s mother came into the den. “What is all the fuss about?” she asked.

“Tell Mary that there isn’t anything that’s impossible!” Peter demanded. “We might not be smart enough to figure it out, but there is always a way to do anything.”

“It is true that we’ve learned how to do a lot of things that people used to think were impossible,” Peter’s mother said. “We can fly faster than the speed of sound; there’s even been a car that drove that fast. We can orbit the Earth, and cure many diseases, and feed billions where millions used to starve.”

“See!” Peter interjected.

“Ah, but maybe there are some things that are impossible,” Peter’s mother continued. “Let me give you an experiment to do on impossibility.”

At that, both Peter and Mary perked up. Doing experiments was one of their favorite activities.

“Let me borrow the chalk and let’s go out to the sidewalk.” Picking up the chalk, Peter’s mother led them all to the sidewalk. “OK, here is the problem. Let’s pretend that the sidewalk is a river. In it are two islands, here and here.” Quickly, she sketched in two large ovals in the middle of the sidewalk. “Now on the big island, there are five bridges. Two go to the east side of the river,” she paused to sketch in two bridges leading to the lawn closest to them. “Two more go to the west side of the river,” she again paused to sketch in two more bridges leading to the lawn on the other side of the sidewalk. “And one goes to the little island. But it also has a bridge going to the east side of the river and another going to the west side.” She finished drawing the last three bridges and stood up. “Now here’s the challenge: Can you walk over all seven bridges without having to walk over any bridge twice? Is it possible to walk over all the bridges just once?”

The bridges

The bridges

“No way!” Mary said. “That’s impossible!”

“There’s no such thing,” Peter insisted. We just haven’t figured it out yet!”

“Well, I’ll be inside working on my exoplanet research,” Peter’s mother said. “Come and get me when you figure it out.”

As she walked into the house, Peter and Mary turned to the sidewalk and started trying different paths.

What do you think will happen? Do the experiment!

 

 

 

 

After about an hour, the two crept back into the house and found Peter’s mother staring intently over light curve data from the latest astronomy satellite.

“I give up,” Peter said. “How do you do it?”

“That’s easy,” his mother replied. “You don’t. This is a famous mathematics problem known as the Königsberg Bridges Problem. You see, in Germany, in a little town called Königsberg, they actually have seven bridges laid out just the way we drew them on the sidewalk. And people used to spend their Sunday afternoons trying to walk over all of the bridges exactly once.”

“This must have been before television,” Mary said.

“Yes, there wasn’t much else to do on Sundays back then. Now, to solve the problem a smart guy by the name of Euler decided that it was much too tiring to walk. Instead, he drew the problem as a bunch of dots connected by lines. There was one dot for the east side, one for the west side and one for each island. And each bridge was a line connecting the dots to each other.”

Peter’s mother turned over a scrap piece of paper and sketched out the diagram.

Euler's solution

Euler’s solution

“Now, Euler simply counted the number of lines leading to each dot. The only way that you can cross all of the bridges only once is if there are no dots with an odd number of lines, or if exactly two dots have an odd number of lines.”

“But we have four dots with an odd number of lines, so that’s impossible.” Peter said.

“Right. The neat thing about this is that highway engineers still use it to help design new interchanges. And computer engineers use it to design computer circuits.”

“So even though it is impossible, it is useful!” Mary said.

“That’s right. And if you’d like a real challenge, spend some time figuring out how many bridges you’d have to add to make the walk possible, and where you’d put them.”

With that, the two headed back out to the sidewalk to build some bridges.

June 1 – Heads for the Hills

Today’s Factismal: While he was a prisoner of war, John Kerrich flipped a coin 10,000 times and got 5,017 heads and 4,983 tails.

You probably didn’t know that today is National Flip A Coin Day, which is  dedicated to improving our understanding of statistics. Though the subject may seem boring, statistics plays an essential part in our lives. From determining if a heat wave is unusual to discovering if the predictions being made by NOAA are worth heeding to simply telling if traffic will be heavier than usual, statistics drives many of our public policy decisions as well as serving as the way we judge the outcomes of experiments. And the amazing thing about statistics is that many of the most powerful statistical tools are based in, you guessed it, coin flips.

But if you’d rather go have a root canal from Orin Scrivello, DDS, than do statistics, then may I suggest an alternative? Panamath is an ongoing citizen science project that is trying to understand our “number sense”, that is, the way that we estimate the number of people in a group or the number of slices of pie left on the table. If you’d like to help, then head on over to
http://panamath.org/

May 17 – It’s a snap

One of the amazing things about science is that what you learn when you study one thing often applies to something else that you wouldn’t think was related at all. In today’s adventure, Daniel, Peter, and Mary discover that spaghetti, earthquakes, animals, and authors all have a secret connection.

 

When Peter’s mother walked into her kitchen, the last thing she expected to see was pieces of spaghetti lying all over the table with Mary and Peter laughing over them. But that’s exactly what she saw. Peter and Mary were sitting on opposite sides of the table, watching each others’ hands very closely as they took turns picking up pieces of spaghetti and snapping them.

“Hey,” Peter’s mother said. “What are you two doing?”

“We’re doing an experiment,” Peter explained. “I read somewhere that spaghetti always breaks into two pieces when you snap it and Mary didn’t believe me. So we decided to see if it was true.”

Looking at the pile of broken spaghetti, his mother shook his head.

“I know that you are supposed to replicate your results, but why so many times? ” she asked. “You should have been able to answer it with the first few pieces.”

“Well,” Peter said. “I can get the spaghetti to break into two pieces pretty consistently but Mary almost always gets three pieces except when she gets four. We’re trying to figure out why.”

“Well, it certainly looks as if you’ve done a pretty thorough job. But have you even decided what the variables are in your experiment?”

“Well, no, ” the two admitted sheepishly.

“So what you are really doing is making a mess,” his mother admonished. “If you don’t know what you’re doing, then you can hardly know if you are doing it right. Tell you what. If you’ll sweep all of this mess into a bowl so we can use it for dinner tonight -”

“Oh, boy! Spaghetti!” Mary interrupted.

“-then I’ll show you a real experiment with spaghetti. Deal?”

By way of agreeing, Peter and Mary quickly jumped up then gathered up the spaghetti pieces into a bowl. Once the table was clear, Peter’s mother put five pieces of whole spaghetti in front of each of them along with a penny.

“Here’s the experiment,” she explained. “What you’ll do is pick up each piece of spaghetti in turn, working from left to right. The other person will flip their coin. If they get heads, then you have to snap the spaghetti into two pieces of equal length by holding it in the middle; you then put the two pieces at the far right side. If the other person gets tails, then you put the piece of spaghetti at the far right and you don’t snap it. Got it?” At their nods, he continued. “You’ll each flip your coin forty times. So you know the variable – the coin flips. Here’s the question: How many pieces of spaghetti will you have at the end?”

“Well, we start with five pieces,” Peter said. “So each one will get eight flips. If they get broken half the time, then each piece should get broken four times giving us two pieces each time and we’ll have five times four times two or forty pieces.”

“I don’t think that’s how it will work,” Mary said. “Remember that this is random. So some of the pieces will get broken more than once and others won’t get broken at all. So it will probably be less than that; I think it will be closer to thirty pieces.”

“Good,” Peter’s mother said. “And notice that you both added an extra prediction. Peter seems to think that the pieces will all be about the same size but Mary, you think that they’ll be different sizes.”

“That’s right,” Peter said.

“Well, there’s only one way to find out,” Peter’s mother said. “Start flipping!”

What do you think will happen? Do the experiment!

For the next ten minutes, the kitchen was filled with the sound of flipping coins and breaking spaghetti. Soon, some of the strands looked less like spaghetti and more like toothpicks while other strands stayed long. At the end of all the flipping, they quickly counted up their pieces of spaghetti.

“Hey! I only got twenty-five pieces!” Peter exclaimed.

“Yes, and one of yours is still whole,” Mary added. “I have thirty pieces but none of mine are longer than a half.”

“So what gives?” Peter asked. “I kept track – we both got the same number of heads.”

“Ah, but did you get them in the same order?” Peter’s mother asked. “Because you kept cycling through the pieces in a strict order, getting heads or tails at a different time meant that you’d get a different length. But, if it makes you feel any better, if you had done more flips on more strands of spaghetti, then you would both have gotten almost identical results. Yours are actually pretty close, given how few times you flipped the coins.”

“How many would we have needed in order to get the same result?” Mary asked.

“An infinite number,” she replied. “But if you’d flipped your coins about 1000 times and used ten strands of spaghetti each, then you two would have been much, much closer.”

“But why didn’t we get forty pieces each?” Peter asked.

“It is just like Mary said – you forgot that you’d be flipping on individual pieces and not the whole strand each time; there’s a fun little math equation that describes what will happen. But what’s really cool about this is that you can do the math backwards and find missing things. Take asteroids for example,” she explained. “We know how many we have found and what their approximate sizes are. By applying that equation, we can predict how many we haven’t found.”

“Cool! Is that why they say that we know where 90% of the big ones are?” Mary asked.

“Yes. But the math doesn’t just apply to rocks in space; it also tells us what the distribution of rocks in an avalanche will look like, and how the number of earthquakes will change with size, and even the number of fish of a given size that are hiding in a lake. Alfred Lotka even used it to predict the number of scientific papers that would be written in any given year.”

“What? How did he do that?” Peter asked.

“He just looked at the number of papers that were written by an scientist who had already published a paper. Let’s say that in one year, there were 500 scientists who each published a paper and that the next year just 125 of them published another paper. Then he knew that in ten years there would only be five of them who wrote a new paper. If you then add up all of the people who were doing research and assumed that they stopped after ten years, you’d just add up the number of scientists publishing for the first time, the second time, and so forth, and discover that there would be 774 papers published that year,” Peter’s mother said. “The neat thing is that it isn’t just papers that you can do that way; it also tells you how emails would be written in a given year and how many blog posts or how many television shows would be filmed.”

“That is so cool!” Mary said. “I didn’t know that one equation could do so much.”

“Well, there is one thing that it can’t do,” Peter’s mother said. “It can’t clear the table so I can make spaghetti!”

With a quick grin, the two hungry young scientists turned to the table and started cleaning up their experiment.

 

 

About the scientist

In 1918, an influenza pandemic swept across the globe and killed one out of every twenty people alive at the time. It was the greatest health disaster since the Black Death and inspired many scientists to find new and better ways of predicting disease outbreaks. One of these was  Alfred Lotka who was trained as both a biologist and a mathematician. He had been working on ways of predicting malaria outbreaks and soon discovered that his work was more widely applicable than he had expected.

He started looking at the number of papers published by scientists, At first, he was just curious about why so many scientists seemed to write fewer papers as they got older but he was surprised when he found that he could actually predict how many papers would be published using a simple formula. His result turned out to be a special case of a more general equation that governs everything from animal populations to gravel size to the number of sunspots.

 

March 29 – Completely Normal

Are scientists normal? In today’s Secret Science Society adventure, Peter and Mary discover what the word means to scientists and why math can be important!

It was another typical Monday afternoon. Peter and Mary had finished lunch and were heading toward their favorite class. They didn’t like science class just because they wanted to be scientists; they also liked the way that their teacher, Mr. Medes, always managed to give them an interesting experiment to do. As a result, they frequently tried to predict what the experiment would be while they ate lunch. Today’s discussion had spilled over and continued as they walked to class.

“Well, it is Monday, so we know that it will be about math,” Mary said.

“Sure, but that doesn’t narrow it down much. Remember last week, when he had us coloring maps all class period?” Peter replied.

“And the week before, when we played with clay,” Mary said as they came to the classroom door. “Well, there’s only one way to find out – let’s go ask Mr. Medes!”

Mr. Medes was waiting for the class as they filed in and found their seats. As soon as all thirty-two students had come in, he started in on his lesson for the day.

“Welcome, welcome! Today is Monday, and that means math! For today’s lesson, we’re going to follow in the footsteps of that great gambler, Pascal. We are going to flip coins!”

“What does gambling have to do with math?” Peter asked.

“Believe it or not, most of the early work in statistics was done by Pascal and other gamblers as they sought a way to make more money. Pascal even invented a gambling machine that we know as the roulette wheel; he did it to see if he could create a perpetual motion machine. Instead, it turned into a perpetual money machine. But we won’t need anything that fancy today. We just need pennies.” As he said that, Mr. Medes passed out a penny to each of the students.

“Now, this is a very simple experiment. What we are going to do is flip the coins. If they are ‘fair coins’, which means if they are evenly balanced, then about half of them should come up heads and half should come up tails. Ready? Everybody flip!”

Suddenly the air was full of flipping coins. When the coins had stopped flipping, Mr. Medes asked for a show of hands from those who had heads.

“OK, we had 17 heads and 15 tails,” he said. “Let’s try that again, just to make sure that wasn’t a fluke.”

They repeated the test four more times, getting about the same distribution of heads and tails each time. Mr. Medes wrote 32 on the board then wrote the last result (14 heads and 18 tails) below it before turning back to the class.

“Good!” Mr. Medes said. “That proves that these are fair coins. We got about half heads and half tails each time, and everyone got a different set of heads and tails. But that raises an interesting question. What do you think would happen if we flipped the coins twice and kept track? How many times would we get either two heads or two tails in a row?”

“Well, you’ve got a fifty-fifty chance of getting heads after a tails, so you’d get it about half the time,” Mary said. “There are thirty two of us, so there’d be sixteen people with two heads or two tails.”

“No, you’re wrong,” Peter said. “You’ve only got a fifty-fifty chance of getting heads or tails on the first flip, so there should only be half as many. We’d have eight people with two heads or two tails. Everyone else will have a head and a tail or a tail and a head.”

“Well, there’s only one way to solve this problem,” Mr. Medes said. “Everyone take out a piece of paper and pencil to record your observations. Flip the coin and write down if it was heads or tails, then do it again.”

What do you think will happen? Do the experiment!

 

 

 

 

For a moment, the classroom was silent as everyone flipped their coins and wrote down the results. As soon as the last penny was put down, Mr. Medes spoke up again.

“OK, how many of you got two heads?” Nine hands went up, and Mr. Medes wrote that on the board. “How many got two tails?” Seven hands went up; Mr. Medes once again recorded the number, then put ‘16’ between them saying “And that leaves 16 people with either a head and a tail or a tail and a head.”

“Where did I go wrong?” Peter asked. “There should have been half that many!”

“Actually, you fell into the same trap that frustrated many of the mathematicians who discovered these rules. They forgot that everyone gets either a head or a tail for their first flip and that the real question is how many people will get a second head or tail. We can write it in math this way:”

Turning to the board, he drew

Second flip

Heads (F) Tails (T)
First Heads (F) FF FT
Tails (T) TF TT

“Hey! That looks like the truth tables my mom uses when she programs!” Peter said.

“That’s right; this branch of math lies at the intersection of programming and statistics. It is also useful in genetics and many other fields. If you count it up, you’ll see that there are two boxes where you get a mixed result and two where you get a ‘pure’ result. So, if you had written it down like this, you’d have known the right answer without needing to do anything more. So – how can we decide how many ‘pure’ results we’ll get if we flip the coins three times?”

“We make another truth table!” Peter said.

“Right,” Mr. Medes replied as he drew a table on the board.

Next flip

Heads (F) Tails (T)
Earlier flips FF FFF FFT
FT FTF FTT
TF TFF TFT
TT TTF TTT

“We’ve got eight possible outcomes, and two of those are ‘pure’. So we should get one eighth of thirty two, or four, people with three heads in a row and the same number of people with three tails in a row; that means we should have eight ‘pure’ results. Now let’s flip the coins and see what happens. After you’ve flipped, raise your hand is you have three in a row of the same, heads or tails.”

Everyone again flipped their coins. The class looked around and burst into smiles as seven people raised their hands.

“Why didn’t we get exactly eight?” Mary asked, with a puzzled look.

“Because this is statistics, which works best for a large number of flips. If we had 320 people flipping the coins, we’d probably get somewhere between 75 and 85 pure results. If we had 3200 people flipping coins, we’d get somewhere between 780 and 820 pure results. As you flip the coin more often, random errors even themselves out and you come closer to the true value. Let’s try an experiment to see why. Mary, if you flip a coin, what are the odds of it coming up heads?”

“It will be heads half the time,” she replied.

“Right. Now Peter, flip your coin and tell me what it shows.”

Peter quickly flipped his coin. “It came up tails!”

“Now that doesn’t mean that Mary was wrong. What it means is that you don’t have enough flips, what a statistician calls ‘a significant population’, to decide what the odds are. The only way to tell the odds is to flip a coin a lot of times. And the more times you flip it, the bigger the population and the less chance that some error has crept into your results. As a matter of fact, a mathematician named John Kerrich tossed a coin 10,000 times while he was being held as a prisoner of war by the Nazis. He got heads 5,067 times and tails 4,933 times. If he had only flipped it once, he would have only gotten a head or a tail, just like Peter did. So bigger is better in statistics.”

At that point, the bell rang and Mr. Medes said “Don’t flip out – I’ll see you tomorrow for more science!”

March 3 – IDIC

Today’s factismal: There are more real numbers between 0 and 1 than there are integers between -∞ and +∞.

Today is Georg Cantor’s 169th birthday; we know this because we can count integers. But it took Cantor to show that there are some things that we couldn’t count and that there’s infinite and really infinite.

Since the days of the Greeks, philosophers have argued over what “infinity” means. Was it just a really big number or was it something even more interesting. In the 1920s, Hilbert described infinity best through paradox; he pointed out that a hotel with an infinite number of rooms would always have room for more guests even if every room was already occupied. Because the rooms are discrete units, you could even (if you had an infinite amount of time) count them all without skipping any; this is what mathematicians call “countably infinite”. And it turns out  that countably infinite is the smallest infinity possible.

The natural numbers (1, 2, 3, 4, … , +∞) are countably infinite. You start at 1 and keep moving forward in regular, discrete intervals until you hit +∞ after an infinite time. And it turns out that the integers are also countably infinite, because you can match every negative integer to a positive one. As a result, the integers, which includes both positive and negative numbers, is exactly as large as the whole numbers, which just has positive integers.

But then things get weird in mathland as the rational numbers, which are defined as of ratios of integers, turns out to be another countably infinite series. To see this, you need to look at Cantor’s clever diagram. He made a square with the numerator across the top (since that’s where numerators belong) and denominators down the side. Because every integer can be matched up with every numerator and every denominator, the total number of rational numbers is just countably infinite.

Numerators
1 2 3 4
Denominators 1 1/1 2/1 3/1 4/1 ∞/1
2 1/2 2/2 3/2 4/2 ∞/2
3 1/3 2/3 3/3 4/3 ∞/3
4… 1/4 2/4 3/4 4/4 ∞/3
1/∞ 2/∞ 3/∞ 4/∞ ∞/∞

So, by applying a little logic and ignoring his common sense, Cantor was able to show that the natural numbers, the whole numbers, the integers, and the rational numbers were all equally big. That would have been enough to satisfy just about anyone except a mathematician. What Cantor and his fellow mathematicians wanted to know was “Is there anything bigger than countably infinite?” And it turns out there there is.

Numerators
1 2 3 4
Denominators 1 0/1 2/1 3/1 4/1 ∞/1
2 1/2 1/2 3/2 4/2 ∞/2
3 1/3 2/3 2/3 4/3 ∞/3
4… 1/4 2/4 3/4 3/4 ∞/4
1/∞ 2/∞ 3/∞ 4/∞ 17/∞

Cantor proved that there were larger infinities by changing the diagram argument slightly. He decided to use the diagonal of the diagram to define a new number. And he defined each diagonal number by saying that it couldn’t be equal to the previous one. As a result, he created a non-repeating, non-rational number; what we call an irrational number. Even better, because the values didn’t match any of the other values, the new number couldn’t be found anywhere in the diagram and so couldn’t be counted. The new number was uncountably infinite. And there were more of these irrational numbers than there were rational numbers. Or, as another mathematician pointed out, there were more real numbers between 0 and 1 than there integers between 0 and ∞.

Though the result seems contradictory, the types of infinity turn out to have some very practical uses. Cantor’s sets have been used to help develop new and better data encryption techniques and to improve computer animation. And we’re still working on new ideas in math, including how our intuitions about numbers can be wrong. If you’d like to help scientists as they try to understand how people think about numbers then head over to Panamath and take their “how many?” test. I promise that the number won’t be infinite and that the test won’t take infinitely long!
http://panamath.org/