A simple economics problem

Photo Credit: Flickr user “Wonderlane” – some rights reserved

Many years to Corbin Foucart for posting “The Simple Math Behind Insurance” for the inspiration. I love simple explanation for economic issues. Some economic things are very complicated. But not all of them. Many years to Corbin for letting simple things be simple. Also, the situation with the die is completely borrowed from him. Thanks for that, too.

Suppose I came to you with a single six-sided die and offered to play a game. Here is how the game goes: If I roll a 6, I will give you $6. If I roll any other number, you have to give me $2. Would you play the game?

I would advise you not to. If we played long enough, I would certainly come out on top. Allow me to explain. For every roll, there is a 1 in 6 chance of getting any of the numbers. Suppose I rolled 36 times. There would likely be about 6 of each number, right? So, with me taking $2 for each of my rolls of numbers 1, 2, 3, 4 and 5, (that is 30 rolls), I would be raking in 60 of your dollars. I still rolled 6 sixes, so I would be giving $36 back. I am still ahead $24. Thus, the game will ultimately end up in my favor.

Now, let’s look at how this model might relate to economics. Consider insurance. We will stick to auto insurance. You know how auto insurance works, right? A person pays monthly to a company that promises to provide the funds to fix (or replace) the car in the event of an accident. You might ask yourself, “How can a company whose business is to pay for other people’s accidents make any money?”

Great question. Answer? The same way that I made money off of you in the game the started this blog post. Even though I was paying out more than I was taking in, I was five times as likely to take money in as I was to pay out. Let’s relate this to insurance.

Suppose you have a small insurance company that insures 100 people.

Each person pays you $25 per month.

The odds, in any given month, of a car getting in a wreck is 1 in 100.

If a car gets in a wreck, you pay out $2000.

This problem is a modified version of the problem posed by Mr. Gordon.

So, each month 100 people pay you $25. So, you have received $2500. Each month, one person gets in a wreck, which costs you $2000.  So, you are ahead $500. You have just become a profitable insurance company. Obviously, there are going to be months where more than one person gets in a wreck, but there will also be months were no one does. That should balance out.

Now, let’s add a variable. Suppose that you know that a certain group of people (we will call them “teenagers”) are more likely to get in a wreck. Let’s also suppose that you just added 25 of those people to your insurance business. How do you change the way you charge your customers?

Suppose the 25 teenagers have a 4 in 100 chance of getting in a wreck.

How do you change your business model? Leave some ideas in the comments. The more specific, the better. (That is, “I’d charge the teenagers more” isn’t as good as it could be. How much more? Why?)

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Good Game, Vol. 5: Sudoku

Photo Credit: Flickr User "Random McRandomhead" - some rights reserved

 

In the spirit of Minesweeper, Sudoku is another game where the game play and rules are take about 30 seconds to explain. Then you begin trying to solve the puzzle.

The game goes like this: You are given a 9X9 square with 81 boxes. You win when you fill in each box with a single one-digit number (1, 2, 3, 4, 5, 6, 7, 8, or 9)  such that each row and column contains no repeats. Also, the 81 squares are divided into 9- 3×3 squares that can’t contain any repeats, either.

That’s it. That’s the whole game. Click here to play a free online version. There are tons of variations as well.

In my opinion, this game flexes the same logic muscles as Minesweeper (see Good Game, Vol. 4) and Armor Logic 2 (see Good Game, Vol. 3). The idea of making-decisions while staying within a small set of parameters is an important step in developing mathematical thinking.

Trucks and SUVs: Collaboration among enemies?

Photo Credit: flickr user “helicon_” – some rights reserved

Photo Credit: flickr user “danietctw” – some rights reserved

Ever wonder why I put so much stock in collaboration in class? Here is a story out of this week’s Detroit Free Press.

Toyota has the market cornered on hybrid vehicles. The Prius alone accounts for over half of the hybrid automobile market in the US. However, ask any farmer why they don’t own a hybrid and the answer will almost always mention that complete lack of towing power.

Ford is known nationwide for their powerful line of “Built Ford Tough” trucks. But, those who have them lament their lousy gas mileage.

The situation is summed up well by Sarah Laskow writing for GOOD in a piece called, “Making Hybrid Pick-ups Work for America.”

“For as long as there have been hybrids, this has been the dynamic. Weenie greenies drive hybrids. People who drive trucks rescue the hybrid drivers when their cars can’t get the job done. But five years from now, the pickup pulling a Prius out of the mud could be a hybrid itself.”

So, what to do about that? Where can an American heavy duty pick-up manufacturer and a Japanese hybrid specialist go to support their respective shortcomings?

How about to each other? According to Greg Gardner, a short-lived international alliance may be on the doorstep where Ford and Toyota work together to create a powerful, yet fuel efficient line of pick-ups and SUVs. (from “Ford, Toyota to collaborate on hybrids” published in the Detroit Free Press on August 23, 2011.)

“Ford is taking advantage of Toyota’s hybrid leadership — it has sold 3.3 million hybrid vehicles since 1997. Toyota will gain from Ford truck-making expertise.”

Here is the bottom line as far as I see it. Collaboration is the way of the future. Don’t think that these two automakers are trying to better the world. Business is still about profit margin. But, they clearly understand that by offering some of their expertise to their competitors and getting some back in return, they can create a product that there is a huge demand for. Creating such a product will benefit their profit margins, even if it helps their competitors as well.

The new economies will be collaborative and so we will learn to be collaborative. It isn’t about reaching higher than those around you. It is about reaching as high as you can. Sometimes that means giving your competitors a boost because you know they can give you one back.

Situations that Cause Division, part II

Photo Credit: Flickr User "rob_rob2001" - Some Rights Reserved

All right, so let’s change the situation a bit. You are your two roommates get up one fall Saturday in college and head to the fridge. You open it up to notice some leftover pizza from the night before. 5 pieces. 3 people. Each wants (and quite frankly expects) an equal share of the pizza pie. So, how do you figure this one out?

A nice mathy solution would be nice, but if you have a practical solution (by practical, I mean not crunching numbers), then offer it and we will try to fasten some number-crunching to model your idea.

Post your ideas in the comments.

Situations that Cause Division, part I

Photo Credit: La Grande Farmer’s Market – Some Rights Reserved

Many years to Christopher Danielson for this one. (I will promo the specific blog post that inspired this at the end… the title would give away to many ideas… anyway.) And, also many years to Mr. Danielson for camping along Lake Superior, which if you haven’t done that, is about the best camping in the midwest… anyway, on to the math.

Suppose you have a recipe for blueberry muffins. The recipe calls for 1/2 cup of fresh blueberries. Suppose further that you actually have 3/4 of a cup of fresh blueberries and you want to use them all. (This, by the way, happened to me this afternoon… I mean it, this exact situation.)

The rest of the ingredient list: 1 3/4 cup flour, 1/3 cup sugar, 1 egg, 2 tsp baking power, 1/4 cup oil, 3/4 cup milk, 1/4 tsp salt.

Remember, you want to use all of the blueberries. You can’t overload your muffins, or they won’t stay together. You must modify your recipe somehow.

How do you do it?

At the end of the last part (this is shaping up to be a three-parter), I will reveal how I modified the recipe to make my muffins.

Post your ideas in the comments.

Good Game, Vol. 4: Minesweeper

Photo Credit: Jim Loy

With the world becoming increasingly Apple based, this wonderful game, which came free on every Windows-bearing computer for the last 20 years could be fading out of style, but allow me to set the record straight: As far as simple logical training goes, this game knows no parallel.

If you have never played before, it goes like this. Each box will be hiding a blank spot, a number or a mine. The object is to identify all of the mines without clicking one, which, of course, blows you up.

If you click a box with a blank spot under it, that means that there aren’t any mines touching that box. If you click a box to reveal a number, that number represents the number of mines in contact with that box. For example, if you click a box and you see a “2”, then of the 8 boxes touching that “2”, 2 of them have mines and 6 do not. The more numbers you reveal, the more clues you get.

If you don’t have a Windows computer, this seems like a pretty good link to play a Java version of the game. Also, I know that the Apple App Store has a very good free download (because I use it) to install on your iDevice.

thegeometryteacher hits 500

photo credit: Public Domain Photos – some rights reserved, used with permission

Milestone #1 complete. thegeometryteacher has surpassed 500 views. This may not seem like a lot for some sites (ESPN.com for example, likely has more than 500 people viewing it at a time just in New York City), but it demonstrates a niche. A community of math thinkers that appreciate interesting and challenging discussion on mathematics… all in the name of learning.

As I reflect on the people that I have met through these last 10 weeks, all by e-mail, (among them: Dan Meyer, Dr. James Tanton, Gaurav Tiwari, Vo Duc Dien, Christopher Danielson, and Jeremy Kun), I am amazed at how many people are here to try to do just what I am doing. (By the way, that list includes people from 4 different countries on three different continents… amazing…) That is, adding their voice to the global conversation demonstrating that there is so much more to math than math classes advertise. This is about us marketing our product, because there is a global benefit to people using it fluently and accurately.

Now, some of the highlights. My first post was May 26th, 2011. This is the 36th post. The most viewed post has been “Ski Jump Luge: Real or Fake?” Followed closely by “A watched pot never boils…” and “The Bowl Problem“. (Speaking of “The Bowl Problem,” many years to Bhanu Pant for being the first person to answer a problem posted in the blog.)

Now, there must be a math problem here somewhere, right? Okay, are you familiar with those unit rate statistics designed to bring awareness to some social malady? “Every 4 seconds worldwide, a person is diagnosed with… [fill-in-the-blank].” Here is my question, which of course will help to bring awareness to thegeometryteacher, fill-in-the-blank: One person visits thegeometryteacher’s blog everyone _________ hours. (on average, of course.)

Cheers, y’all. I will update the statistics if we ever make it to 1000 views. Much love for all of the support and I will try my best not to let you down moving forward.