The Challenge
Probability and statistics can be a real challenge for students who can see the topics as opaque, and at times unintuitive.
As teachers, we use familiar games of chance to teach some of the fundamental ideas of probability. A coin flip becomes the perfect metaphore for 50/50 split, 2 equally possible outcomes. Dice add some complexity with its 6 possible sides. These probabilities, when combined, provide the foundation for understanding basic probabilistic principles which extend all the way into advanced robotics.
The challenge with coins and dice is that it takes some imagination (or some work) to actually see the probability patterns. For example, a young person might play with dice for years and years, but when shown the probability distribution of rolling a pair of dice, they are likely to draw a blank stare.
This is where the Aakkozzll comes in.
The App
The Aakkozzll on the right is web-enabled. »
Click on the TAP TO DROP button to drop balls.
Originally created in the form of a coin, and now available digitally for web, iPhone, iPod, and iPad, the Aakkozzll was designed to be used as a substitute for dice. When you tap to drop, 1024 little balls are released and tumble down into slots numbered 2 to 12, providing the same values as the roll of a pair of dice.
Of course the distribution is much different. The chances of rolling a 2 with a pair of dice is 1/36, whereas it is 1/1024 with the Aakkozzll.
By playing with the Aakkozzll, our hope is that young people gain a new intuitionfor probability, and become familiar with the normal curve from a young age. The games you can play involving the Aakkozzll are completely unthreatening and don't feel like math class. But, the insights they will use to win will develop a powerful intuition for probability and statistics.
Click on the next tab above to see some games you can play with the Aakkozzll.
Craps
Humans have been playing variations of this game for thousands of years, and it continues to deliver a tremendous amount of excitement for all who play. There is nothing like learning a new skill or insight when you have something on the line.
Learn the Rules
Setup
You need 2 to 8 players, an Aakkozzll (iPhone, Web, or you can use a pair of dice). The game is fun to play with pennies or betting chips! Otherwise, you can always keep track of the score on a piece of paper.
Rules
- Decide who rolls first. Each player can take turns rolling the Aakkozzll, and the highest roller gets to shoot first. The shooter is the person who rolls the Aakkozzll.
- The shooter makes the first bet. The bet is that he/she will win (there are a few ways to win as we will see later). Then he goes around the group and collects counter-bets. This determines who gets what if the shooter loses.
- Once the betting is done, the shooter taps the Aakkozzll to roll the come out.
- If the shooter rolls a 7 or an 11, he wins the bet! The betting starts again!
- If the shooter rolls a 2, 3, or 12, he loses the bet! The Aakkozzll goes to the next player (going clockwise).
- If the shooter rolls anything else (4, 5, 6, 8, 9, 10), then this becomes the point number and the shooter gets to keep rolling.
- If the shooter rolled a point number, he will need to keep rolling until he gets that number again. If he does, he wins (and he gets to bet again)! But, if he rolls a 7, he loses and passes the roll on to the next player.
- With every bet, you either double what you put in, or you lose it. To make it easy to manage the bets, etiher write them on a piece of paper (who's betting against who, how much), or line up your chips in paired stacks.
Pedagogy
The key insight comes when players realize that 7 is the most likely roll. The odds are for them in their first roll (where a 7 gives them a win, and the highly unlikelt 2,3, and 12 makes them lose), but turn against them if they hit point (then the most probable 7 becomes a losing roll). The odds are not obvious when using a pair of dice, but they are visible on every roll of the Aakkozzll (as long as players make the connection that the colored ball has the same odds as all of the others).
Over time, players gain an intuition for how unlikely events in the tail are in normal curve. Basic human psychology tends to react most dramatically over unlikely events (winning the lottery). In this context, players are always constantly aware of just how lucky they are.
Liar
This exciting and playful game is bound to make a conniver out of you! This game requires the ability to keep the result of the roll secret so is recommended primarily for mobile versions of the Aakkozzll.
Learn the Rules
Setup
You need at least 2 players, an Aakkozzll (iPhone recommended). It's good to keep the sound on to prvent cheating. You can optionally keep track of winners and losers on a piece of paper.
Rules
- The first player rolls (keeping the result hidden).
- He tells the person to his right what he rolled (without showing). The next person can either accept the challenge and try to roll something higher, or can call the roller a liar.
- If roller is called a liar, he/she has to show the true result of the roll. If it was a lie, then he loses and is out of the game. If he/she was telling the truth then the person who accused him of lying is out of the game, and the Aakkozzll is passed on to the next player as the game continues.
- If the player doeesn't call the previous roller a liar, he/she needs to roll something equal to or higher than what the previous person said.
- But now the next player has an opportunity to lie. Even if he doesn't roll something higher, he/she can still say that he did, and then its up to the next person on the left to either accept the challenge or accuse him/er of lying.
- This goes on, going around the circle until only one person is left.
To summarize, people fall out in one of 2 ways:
- They are caught lying.
- They admit rolling something lower than the previous roll.
Note that the person who rolls first doesn't lose if their role is beaten, that's just the way goes on.
Pedagogy
There are three elements to understanding how to play this game well.
#1. What are the chances that you will roll a better score?
#2. What are the chances that the previous person is lying?
#3. What are the chances that you will be caught lying?
These chances are all inter-related. If your chances of rolling a better score are low, and you say that you did, then you are probably lying, and if someone knows this, they will pobably catch you. How do you know if the chances of beating the roll are low? Beating a 7 can be a coin toss, but beating an 11 is a one in a thousand occurence.
Luckily, the probability of each roll is quite visible to each player, it is just a matter of understanding what frequencies are!
Create your Own Game
The Aakkozzll might be used as a replacement for almost any game traditionally played with dice. Some games are more fun than others, but the best game is the one you design yourself! If you have an orginal (or classic) game idea you would like to share with others, fill out the form below an you might see your idea on this page.
The Quincunx (pronounced quinn-cux) or bead board, as some call it, was developed by a mathematician named Galton in 1873 (complete history noted below). The device works by dropping a series of acrylic balls, or beads, through rows of precisely located pins. Each bead, as it hits a pin, has a 50-50 chance of falling to the left or right. Each bead then continues to fall over subsequent rows of pins and eventually lands in a slot or cell. The shape of the accumulated beads in the cells forms a pattern or what is statistically referred to as a bell shaped or normal distribution.
Read more history from Jim Warren's page
The Normal Curve later went on to revolutionize science, creating the basis for quality control, medical trials, modern investment strategies, and a revolution in the social sciences.
The Normal Curve occurs everywhere in nature, it is as ubiquitous as the circle. Yet, few have the opportunity to really notice it, as it is hidden in frequencies, buried in numbers.
Burke Brown created the Aakkozzll coin as a toy which children as young as 5 could play with and thereby gain commerce with the curve. The Aakkozzll embodies the same physical elements as the galton board with a very minor modification. One of the balls is colored (differentiating it from all of the others) and the slot that this colored ball falls into determines the value of the roll.
In an attempt to get this into as many hands as possible, the coin was later converted into a web/mobile app which is distributed completely free of charge across the world. It is up to inspired and motivated teachers to put this device in as many young hands as possible so that they may all experience the miracle of the curve.
Aakkozzll for Android
Coming Soon
Aakkozzll Coin
Coming Soon
Aakkozzll T-Shirt
Coming Soon