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. After 2n plays he has won exactly n times. Has he more 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. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

From A Random World to a Rational Universe

Stage: 2, 3 and 4

Published June 2008,February 2011.

Pedagogical Notes

1. Do pupils have sufficient experience of Dice games? Do they have any intuitive beliefs about which throws are more or less frequent? Do they still play 'Snakes and Ladders', 'Ludo' or 'Monopoly'? Do they use dice in 'Role Playing' games like 'Dungeons and Dragons'?
2. Computer simulations exist, but do they provide a genuine 'feel' for the fall of the dice, and what experience do pupils have of throws of two or three dice?

3. We still talk about 'Luck' in getting advantageous results when throwing dice. Pupils can make up a double-entry dice table for two dice (e.g. red die along the top and blue die along the side). This can be used to explain the number of possible combinations for particular throws from 2 to 12. How would you construct a table for three dice?

4. What activities exist in other cultures (Asian, Chinese, etc.) where dice or other objects are used for making choices in rituals or games?

5. In Primary school dice are often marked with numbers or arithmetic operations and used for specially designed board games. What opportunities are there for using more than one die?

6. Dice can be made in the shapes of the Platonic solids. Consider the odds for different polyhedral dice.

7. Count the number of throws for three dice as shown in the table in the poem De Vetula. Notice that the written numerals on the left give a different number of throws. Compare this with Galileo's table of results. How do pupils explain the difference in these results?

8. 3 x 3 Magic Squares are well known in school, but do we ever investigate higher order magic squares? Make a list of the line-totals (34, 65, 111, ...) of the magic squares up to the 'Square of the Moon'. Now make a difference table for this list. What do you notice? How can you explain the result? Now investigate the sums of all the numbers in each Magic Square.

9. Pascal's Triangle can be used for calculating the odds with two dice. Is there a similar polynomial expansion for use with three dice