Designing a Game
Introduction
By the end of the lesson, you will hopefully be able to design a game and calculate the relevant probabilities to decide on whether it is a fair game or not. Let's begin with a few games first:
Game 1
Roll two 4-sided dice, multiply the numbers
You get $0 if you roll a 2 or 4
You get $2 if you roll a multiple of 3
You get $5 if you roll a 8 or 16
You get $10 if you roll a 1
How would you work out the probability of the above events?
What would be the expected return of the game?
Was it a fair game? What would you charge to make this a fair game?
What is the relative frequency/experimental probability for playing this game?
Use this worksheet to help you understand all the concepts above.
Game 2
Roll 2 dice, add the numbers up
You get $1 if you get a prime
You give me $1 if you don’t
We play it 36 times
Who wants to gamble with me?
What is the relative frequency/experimental probability?
Draw a sample space, what is the theoretical probability?
What is the expected value for playing 36 games? What is the expected value for playing one game?
Was it a fair game? How can you make it fair?
Game 3
Roll 2 dice, multiply the numbers up
You get $2 if you get an odd
You give me $1 if you don’t
We play it 36 times
Who wants to gamble with me?
What is the relative frequency/experimental probability?
Draw a sample space, what is the theoretical probability?
What is the expected value for playing 36 games? What is the expected value for playing one game?
Was it a fair game? How can you make it fair?
Game 4
2 players
Rock, Paper, Scissors
You pay $5 to play
You get $6 if we play different things e.g. R and S
You get $3 if we play the same thing e.g. R and R
We play it 36 times
Who wants to gamble with me?
What is the relative frequency/experimental probability?
Draw a sample space, what is the theoretical probability?
What is the expected value for playing 36 games? What is the expected value for playing one game?
Was it a fair game? How can you make it fair?
Game 5
3 players
Rock, Paper, Scissors
You pay $5 each to play
You get $18 each if we all play the same things e.g. R, R and R
You get $9 each if we all play different things e.g. R, P and S
You get $3 each if two of us play the same thing e.g. R, R and S
We play it 36 times
Who wants to gamble with me?
What is the relative frequency/experimental probability?
Draw a sample space, what is the theoretical probability?
What is the expected value for playing 36 games? What is the expected value for playing one game?
Was it a fair game? How can you make it fair?
Now you design your own game
You can:
Roll at least 2 dice
Spin at least 2 coins
Roll 1 die and toss 1 coin
Pick up to 2 cards from from 4Js 4Qs 4Ks
You can charge up to $10
Write down the rules, including how much each pays and how much they get
Calculate the relative frequency/experimental probability
Draw a sample space, and calculate the theoretical probability
Calculate the expectation
Show whether it is a fair game or not and make it fair if it isn’t
Further Questions and Challenges
What makes a good game in terms of the win/loss ratio for a potential player?
What’s good for the bank?
What makes a popular game in terms of the odds?
Extension
You may wish to take a look at this activity, an experiment conducted by two investment fund managers Victor Haghani and Rich Dewey. I the experiement, they invited 61 people, a combination of college-age students in finance and economics and some young professionals at finance firms (including 14 who worked for fund managers), to take a test. They were each given a stake of $25 and then asked to bet on a coin that would land heads 60% of the time. The prizes were real, although capped at $250. What would your strategy be?
Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment. Neither approach is in the least bit optimal. The best strategy was devised by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956. If you are interested, google it!