Coloured Cubes

Introduction

The following game is based on this "In a box" task on nRich:

  • Chris and Jo put some red and some blue cubes in a box.

  • They each pick a cube from the box without looking (and without replacing them).

  • Jo wins if the two cubes are the same colour and Chris wins if the two cubes are a different colour.

  1. How many cubes of each colour would you need in the box to make it a fair game?

  2. Is there more than one way to make a fair game?


Support

You may wish to explore the questions above or take a look at the following prompts:

  • Start small, how many red and blue cubes would you start with?

  • How would you systematically change the number of cubes to help you see patterns more easily?

  • How would you draw sample space diagrams to show your results? Is there another way to present your findings?

  • Play the game a few times, what are the experimental probabilities for Jo and Chris to win?

  • Is this a fair game? How can we be sure?

  • How many goes do you think we need to be confident of the likelihood of winning?

  • Can you justify your general rules/ conclusions for different numbers of cubes?

Further Questions and Challenges

What would happen if you had colours of 3 different colours? 4? n?

Extension

Take a look at these similar tasks from nRich, they get more and more difficult:

Win or Lose - a gambler bets half the money in his pocket on the toss of a coin, winning an equal amount for a head and losing his money if the result is a tail. He repeats this many times, each time betting half the total money he has. After 2n plays he has won exactly n times. Has he more money, the same amount or less money than he started with?


Fixing the Odds - You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. Your friend chooses a bag at random and then chooses a ball at random from that bag. How should you distribute the balls between the two bags so as to make the probability that your friend will choose a red ball as small as possible and what will the probability be in that case?

How should you distribute the balls so as to make the probability of choosing a red ball as large as possible and what will the probability be in that case?

What happens if you have two bags, a hundred red balls and a hundred white balls?


Scratch Cards - Mail order companies often send a scratch card with their catalogue. These cards are used to encourage people to order from the catalogue by offering a free prize. You have to uncover three of the numbers by scratching off the cover and revealing the number under three of the circles. Of course the numbers under the circles are randomly placed. If the numbers add up to more than fifteen you win a prize. What is the probability of winning a prize?